Part 1 – Probability and Distribution Theory
Statistical Inference and Regression
Analysis:
Stat-GB.3302.30, Stat-UB.0015.01
Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
Part 1 – Probability and Distribution Theory
Part 1 –Probability and
Distribution Theory
Part 1 – Probability and Distribution Theory
1 – Probability
Part 1 – Probability and Distribution Theory
4/108
Sample Space
Random outcomes: The result of a process




Outcomes, experiments and sample spaces
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Frequency
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Listing
Pie Chart of Percent vs Type
Percent

Mushroom and Onion
9.2%
Sequence of events,
Number of events,
Measurement of a length of time, space, etc.
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
5/108
Consumer Choice: 4 possible ways a
randomly chosen traveler might travel
between Sydney and Melbourne
 = {Air, Train, Bus, Car}
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
6/108
Market Behavior: Fair Isaacs credit card
service to major vendors
 = {Reject, Accept}
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
7/108
Measurement of Lifetimes

A box of light bulbs states “Average life is 1500
hours”

Outcome = length of time until failure (lifetime) of a
randomly chosen light bulb
 = {lifetime | lifetime > 0}
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
8/108
Events


Events are defined as
 Subsets of sample space, such as empty set
 Intersection of related events
 Complements such as “A” and “not A”
 Disjoint sets such as (train,bus),(air,car)
Any subset including  is a disjoint union of
subsets:
 = (Air, Train)  (Bus, Car)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
9/108
Probability is a Measure
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Percent
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Listing

Percent

The sample space  is a  - field:
 Contains at least one nonempty subset (event)
 Is closed under complementarity
 Is closed under countable union
Probability is a measure defined on all subsets of 
Axioms of Probability
 P() = 1
 A    P(A) > 0
 If A  B = {}, P(A  B) = P(A) + P(B)
Listing

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
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50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
10/108
Implications of the Axioms
P(~A) = 1 – P(A) as A  ~A = 
 P() = 0 as  = ~  and P() = 1

A  B  P(A) < P(B) as B = A + (~A  B)
 P(A  B) = P(A) + P(B) – P(A  B)

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
11/108
Probability
Assigning probability: ‘Size’ of an event relative to
size of sample space.
Counting rules for equally likely discrete outcomes
 Using combinations and permutations to count
elements
 Example: Discrete uniform, poker hands
 Example hypergeometric: the super committee
(House 242R,193D, Senate 49R, 51D&I)
Measurement for continuous outcomes
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
12/108
Applications:
Games of Chance; Poker
In a 5 card hand from a deck of 52,
there are (52*51*50*49*48)/(5*4*3*2*1)
different possible hands. (Order
doesn’t matter). 2,598,960 possible
hands.
 How many of these hands have 4
aces? 48 = the 4 aces plus any of the
remaining 48 cards.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
13/108
Some Poker Hands
Full House – 3 of one kind, 2 of another.
(Also called a “boat.”)
Royal Flush – Top 5 cards in a suit
Flush – 5 cards in a suit, not sequential
Straight Flush – 5 sequential cards in the
same suit suit
Straight – 5 cards in a numerical row, not the same
suit
4 of a kind – plus any other card
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
14/108
5 Card Poker Hands
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
15/108
The Dead Man’s Hand
The dead man’s hand is 5 cards, 2 aces, 2 8’s
and some other 5th card (Wild Bill Hickok was
holding this hand when he was shot in the back
and killed in 1876.) The number of hands with
two aces and two 8’s is  4   4  44 = 1,584

2 2
The rest of the story claims that Hickok held all
black cards (the bullets). The probability for this
hand falls to only 44/2598960. (The four cards in
the picture and one of the remaining 44.)
Some claims have been made about the 5th card,
but noone is sure – there is no record.


http://en.wikipedia.org/wiki/Dead_man's_hand
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
16/108
Budget Supercommittee
D
 57 
Senate  
3
R
I
 41
 
3
 2
 
 0
How many possible committees? About 9.3E20
57(56)(55) 41(40)(39) 2(1)

3(2)(1)
3(2)(1) 2(1)
193   242   0 
House 
 
  
 3   3   0
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
193(192)(191) 242(241)(240) 1
3(2)(1)
3(2)(1)
1
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
17/108
Conditional Probability
P(A|B) = P(A,B)/P(B)
= Size of A relative to a subset of 
 Basic result p(A,B) = p(A|B) p(B)
(follows from the definition)
 Bayes theorem

Applications – mammography, drug
testing, lie detector test, PSA test.
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
18/108
Using Conditional Probabilities: Bayes
Theorem
P(A,B)
P(A | B) 
P(B)
P(B | A)P(A)

P(B)
P(B | A)P(A)

P(A,B)  P(notA,B)
P(B | A)P(A)

P(B | A)P(A)  P(B | notA)P(notA)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
100000
15000
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
300000
10
Mean
StDev
N
10
500000
400000
20
300000
200000
60
50
40
30
Normal
100
12
700000
600000
70
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
80
600000
200000
369687
156865
51
0.994
0.012
Computation
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
Definition
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Theorem
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Target
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
19/108
Drug Testing
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
What are P(D|+) and P(N|–)?
Note, P(D|+) = the probability that a patient actually has the
disease when the test says they do.
Percent

Frequency

Listing

Data
 P(Test correctly indicates disease)=.98 (Sensitivity)
 P(Test correctly indicates absence)=.95 (Specificity)
 P(Disease) = .005 (Fairly rare)
Notation
 + = test indicates disease, – = indicates no disease
 D = presence of disease, N = absence of disease
Data:
 P(D) = .005 (Incidence of the disease)
 P(+|D) = .98
(Correct detection of the disease)
 P(–|N) = .95
(Correct failure to detect the disease)
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
20/108
More Information
Deduce: Since P(+|D)=.98,
we know
P(–|D)=.02
because
P(-|D)+P(+|D)=1
[P(–|D) is the P(False negative).
 Deduce: Since P(–|N)=.95,
we know
P(+|N)=.05
because
P(-|N)+P(+|N)=1
[P(+|N) is the P(False positive).
 Deduce: Since P(D)=.005,
P(N)=.995
because
P(D)+P(N)=1.

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
21/108
Now, Use Bayes Theorem
We have P(+|D)=.98.
What is P(D|+)?
P(D and +)
P(+|D)P(D)
=
(By Bayes Theorem)
P(+)
P(+)
P(+) = P(D and +) + P(N and +)
P(D|+)=
= P(+|D)P(D) + P(+|N)P(N) so
P(D|+) =
=
P(+|D)P(D)
P(+|D)P(D)
=
P(+)
P(+|D)P(D) + P(+|N)P(N)
.98(.005)
= 0.08966 (!!)
.98(.005)+.05(.995)
Using the same approach, P(N|-) = 0.999889
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
22/108
Independent events
Plain
32.5%
800000
800000
500000
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
900000
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
Application: Infectious disease
transmission
Frequency
Sausage
5.8%

Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
Multiplication rule P(A,B) = P(A)P(B)
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Definition: P(A|B) = P(A)
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
2 – Random Variables
Part 1 – Probability and Distribution Theory
24/108
Random Variable
Definition: Maps elements of the
sample space to a single variable:
 Assigns a number to   

Discrete:
Payoff to poker hands
 Continuous: Lightbulb lifetimes
 Mixed:
Ticket sales with capacity
constraints. (Censoring)

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
25/108
Market Behavior: Fair Isaacs credit card
service to major vendors
= {Reject, Accept}
X = 0=reject, 1=accept
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
26/108
Caribbean Stud Poker
{---------------- Sample Space --------------}
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
Variable
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Probability
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
27/108
Features of Random Variables

Probability Distribution
Mass function:
Prob(X=x)=f(x)
 Density function: f(x), x = ...


Cumulative probabilities; CDF
Prob(X < x)
 F(x)

Quantiles: x such that F(x) = Q
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Median: x = median, Q = 0.5.
Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
28/108
Discrete Random Variables
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Frequency

Listing

Elemental building block
 Bernoulli: Credit card applications
 Discrete uniform: Die toss
Counting Rules
 Binomial: Family composition
 Hypergeometric: House/Senate Supercommittee
Models
 Poisson: Diabetes incidence, Accidents, etc.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
29/108
Market Behavior: Fair Isaacs credit card
service to major vendors
X = 0=reject, 1=accept
Prob(X=x)=(1-p)(1-x)px, x=0,1
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
30/108
Binomial
n x
n-x
Prob(X=x)=   p (1-p)
x
Sum of n Bernoulli trials
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
31/108
Examples
Family has 4 children.
 4 1 
Prob[4 daughters] =   
 4 2 
4
0
1
 1
1




 2  16
Weapons system has 20 components. It will fail if 2 or more break down.
Prob any component fails is 0.2
Prob(system breaks down) = Prob[X  2] =

 20  2 20 x
  .2 .8
 x
20
x2
= 1 - Prob[X < 2]
= Prob(X=0) + Prob(X=1)
= (1).2 0 .820 + (20).21 .819  .819 (.8  (20).2))  .069167
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
32/108
Poisson
800000
800000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
Frequency
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
General model for a type of process
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Approximation to binomial
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
33/108
Poisson Approximation to
Binomial
n
Prob(X=x|n,p)=   p x (1  p) n  x
 x
 
1  
 n
n x
,  = np
  x 
  
n!
=   x
 1  
 x !   n (n  x)!   n 
exp( ) x


, x = 0,...,n
n 
x!
Approximation improves as  
 0
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
4
200000
Normal
10
500000
300000
0
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
100
12
700000
400000
10
17500
Histogram of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000

 e

1
900000
Mean
StDev
N
AD
P-Value
95
600000
x
Scatterplot of Listing vs IncomePC
Normal - 95% CI
99
700000
300000
100000
Probability Plot of Listing
 
1  
 n
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
n

1
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
x
Percent
n!


 
x !(n  x)!  n 
  np, n  ,p 
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
34/108
Diabetes Incidence per 1000
exp(-7)7d
Prob(D=d) 
, d=0,1,...,1000.
d!
http://www.cdc.gov/diabetes/statistics/incidence/fig2.htm
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
35/108
Poisson Distribution of Disease
Cases in 1000 Draws with =7
Poisson Probabilities for Diabetes Cases
0.16
0.14
PoissonProbability
0.12
0.10
0.08
0.06
0.04
0.02
0.00
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
Mean
StDev
N
369687
156865
51
80
8
4
200000
Normal
10
500000
300000
0
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
100
12
700000
400000
10
16
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
14
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
12
Scatterplot of Listing vs IncomePC
Normal - 95% CI
99
700000
300000
100000
Probability Plot of Listing
10
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
8
Cases
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
6
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
4
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
2
Frequency
0
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
36/108
Poisson Process: Doctor visits in the survey year
by people in a sample of 27,326.  = .8
Poisson probability model is a description of this process, not an approximation
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
37/108
Continuous RV
Plain
32.5%
800000
800000
500000
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
900000
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
Application: Lightbulb lifetimes?
Frequency
Sausage
5.8%

Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
Probability measure P(event)
obtained using the density.
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Density function, f(x)
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
38/108
Probability Density Function; PDF
Density: f(x)  0

Definition:

f(x)dx=1
-
Definition: CDF=Prob(X  x)=F(x)= 
x
-
f(x)dx
('Cumulative Density Function' or 'Distribution Function')
Probability: P(ω)=  f(x)dx
ω
b
In range a to b Prob(a < X < b) = Prob(a  X  b)   f(x)dx
a
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
39/108
CDF and Quantiles

pth = quantile; 0 < p < 1

Quantile = xp such that F(xp) = p.
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
For p = .5, xp = median
Listing

xp = F-1(p).
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
40/108
Model for Light Bulb Lifetimes
This is the exponential model for lifetimes. The model is f(time) = (1/μ) e-time/μ
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
41/108
Model for Light Bulb Lifetimes
The area under the entire curve is 1.0.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
42/108
Continuous Distribution
The probability associated with an interval such as 1000 < LIFETIME < 2000 equals the area under the curve
from the lower limit to the upper.
A partial area will be between
0.0 and 1.0, and will produce a
probability.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
43/108
Probability of a Single Value Is Zero
The probability associated with a single point, such as LIFETIME=2000, equals 0.0.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
44/108
Probabilities via the CDF
b
Prob(a  X  b)   f ( x)dx
a
 F (b)  F (a)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
45/108
Probability for a Range of Values Based on CDF
Prob(Life < 2000)
(.7364)
Minus
Prob(Life < 1000)
(.4866)
Equals
Prob(1000 < Life < 2000) (.2498)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
46/108
Common Continuous RVs
Continuous random variables are all models; they do
not occur in nature. The model builder’s toolkit:
 Continuous uniform
 Exponential
 Normal
 Lognormal
 Gamma
 Beta
Defined for specific types of outcomes
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Listing

Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
47/108
Continuous Uniform
800000
800000
500000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
Frequency
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
F(x) = x/(b – a), a < x < b.
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
f(x) = 1/(b – a), a < x < b
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
48/108
Exponential
f(x) =  exp(-x), x > 0, 0 otherwise
 F(x) = 1 – exp(-x), x > 0

Median: F(M) = .5
1 – exp(-M) = .5
exp(-M) = .5
– M = ln.5
M = -ln.5/
= (ln2)/
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
49/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
49
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
50/108
Gamma Density Uses the
Gamma Function
Definition
( P)  

0
x P 1e  x dx (Note the limits of integration.)
Extremely Useful Recurrence (Prove by integration by parts)
( P )  ( P  1)( P  1)
Implication for positive integers: ( P)  ( P  1)! and (1)  0!  1
Also useful special case: (1 / 2)  
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
51/108
Gamma Distributed Random Variable
 P x P 1ex
f ( x | , P ) 
, x  0,  > 0, P > 0.
( P )
Used to model nonnegative random
variables – e.g., survival of people and
electronic components
Two special cases
P = 1 is the exponential distribution
P = ½ and  = ½ is the chi squared
with one “degree of freedom”
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
52/108
Beta Uses Beta Integrals
Beta density is used to model a random variable that ranges
from 0 to 1 such as a proportion
1
(a, b)   x a 1 (1  x)b 1 dx 
0
 ( a )  (b )
( a  b)
( a  b) a 1
x (1  x)b 1 ,
 ( a )  (b )
0  x  1, a > 0, b > 0.
The beta density is f(x|a,b) =
Useful special case, a = 1 and b = 1 is the Uniform(0,1)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
53/108
Normal Density – The Model
 1 x -μ
f(x) =
exp - 

σ 2π
 2  σ 
1
2

, -  < x < + 

Mean = μ, standard deviation = σ
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
54/108
Normal Distributions
The scale and location
(on the horizontal
axis) depend on μ and
σ. The shape of the
distribution is always
the same. (Bell curve)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
55/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
56/108
Standard Normal Density (0,1)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
57/108
Lognormal Distribution
 (ln x  )2 
f ( x) 
exp 
 , x > 0.
2
2
x 2


1
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
58/108
Censoring and Truncation
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Frequency
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Listing

Censoring
 Observation mechanism. Values above or below a certain
value are assigned the boundary value
 Applications, ticket market: demand vs. sales given capacity
constraints; top coded income data
Truncation
 Observation mechanism. The relevant distribution only
applies in a restricted range of the random variable
 Application: On site survey for recreation visits. Truncated
Poisson
 Incidental truncation: Income is observed only for those
whose wealth (not income) exceeds $100,000.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
59/108
Truncated Random Variable
Untruncated variable has density f(x)
 Truncated variable has density
f(x)/Prob(x is in range)
f(x)
 Truncated Normal: f(x|x  a)=

Prob(x  a)
 σ
1
=
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Marginal Plot of Listing vs IncomePC
Normal
100
12
700000
400000
10

,x  a
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
σ
σ
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000

1-Φ (a-μ)
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

 (x-μ)
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
60/108
Truncated Normal:
f(x|x>a) = f(x)/Prob(x>a)
F(x | x > XL )
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
61/108
Truncated Poisson


f(x)= exp(-) x / (x+1)
f(x|x>0) = f(x)/Prob(x>0)
= f(x) / [1 – Prob(x=0)]
= {exp(-) x / (x+1)} / {1 - exp(-)}
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
62/108
Representations of a Continuous
Random Variable

Representations
Density, f(x)
 CDF, F(x) = Prob(X < x)
 Survival, S(x) = Prob(X > x) = 1-F(x)
 Hazard function, h(x) = -dlnS(x)/dx

800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Frequency
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Representations are one to one –
each uniquely determines the
distribution of the random variable
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
63/108
Application: A Memoryless Process
Hazard Rate
h(t )  lim t 
 0 Prob(t < T < t+t|T  t)
= Probability of failure in the next t interval given that
failure has not occurred up to t.
In a memoryless process, the hazard rate does not depend on how
long the process has been ongoing.
h(t) =  (a constant)
d ln S (t )
 .
dt
General solution: S(t) = Kexp(-t)
Terminal condition: S(0) = 1, so K = 1 and S(t) = exp(-t)
h(t)= -
F(t) = 1 - S(t) = 1 - exp(-t) = the CDF of the exponential random variable
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
64/108
A Change of Variable
Theorem: x = a continuous RV with
continuous density f(x). y=g(x) is a
monotonic function over the range of x.
y=g(x), f(y) = f(x(y)) |dx(y)/dy)|
= f(x(y)) |dg-1(y)/dy)|
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
65/108
Change of Variable Applications
Plain
32.5%
800000
800000
500000
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
600000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
900000
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
Fundamental probability transform
Frequency
Sausage
5.8%

Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
Lognormal to normal
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
Standardized normal
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
66/108
Standardized Normal
Pie Chart of Percent vs Type
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
Mushroom and Onion
9.2%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing

Percent

Frequency

Listing

X ~ N[, 2]
Prob[X < a] = F(a)
Prob[X < a] = Prob[(X - )/ ] < (a - )/
y = (x - )/ 
 J = dx(y)/dy = 
 f(y) = f(y+ )   = [1/sqr(2)]exp(-y2/2)
Only a table for the standard normal is needed.
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
67/108
Textbooks Provide Tables of
Areas for the Standard Normal
Econometric Analysis, WHG, 2008,
Appendix G, page 1093, Rice Table 2
Note that values are only given for z
ranging from 0.00 to 3.99. No values
are given for negative z.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
68/108
Computing Probabilities
Standard Normal Tables give probabilities
when μ = 0 and σ = 1.
 For other cases, do we need another
table?
 Probabilities for other cases are obtained
by “standardizing.”

Standardized variable is z = (x – μ)/ σ
 z has mean 0 and standard deviation 1

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
69/108
Standard Normal Density
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
70/108
Standard Normal Distribution Facts
The random variable z runs from -∞ to +∞
 (z) > 0 for all z, but for |z| > 4, it is
essentially 0.
 The total area under the curve equals 1.0.
 The curve is symmetric around 0. (The
normal distribution generally is symmetric
around μ.)

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
71/108
Only Half the Table Is Needed
The area to left of 0.0 is exactly 0.5.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
72/108
Only Half the Table Is Needed
The area left of 1.60 is exactly 0.5 plus the area between 0.0 and 1.60.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
73/108
Areas Left of Negative Z
Area left of -1.6 equals area right of +1.6.
Area right of +1.6 equals 1 – area to the left of +1.6.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
74/108
Computing Probabilities by Standardizing:
Example
P  4.5  x  8 |   3.5,   2.0 
8  
x
 4.5  


P
 

 
8  3.5 
x  3.5
 4.5  3.5


P
2.0 
2.0
 2.0
 P[0.5  z  2.25]
 P[z  2.25] - P[z  0.5]
 0.9878  0.6915
 0.2963
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
75/108
Lognormal Distribution
 (lnx-μ) 2 
f(x)=
exp  , x > 0.
2
2σ
xσ 2π


1
y = lnx, x = exp(y), dx/dy = exp(y)
 ( y  ) 2 
f(y) =
exp  
 exp( y )
2
2 
exp( y ) 2

1
 ( y  ) 2 
1

exp  
.
2
2 
 2

If x has a lognormal distribution, ln x has a normal distribution
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
76/108
Lognormal Distribution of
Monthly Wages in NLS
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
76
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
77/108
Log of Lognormal Variable
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
77
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
78/108
Fundamental Probability
Transformation
x ~ f(x). F(x) is the cdf of x.
What is the distribution of y = F(x), 0  y  1?
The CDF of y is F(y).
Prob(Y  y) = Prob(F(x)  y)
= Prob(x  F-1 (y)]
= F(F-1 (y))
= y
This is a uniform distribution.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
79/108
Random Number
Generation
The CDF is a monotonic function of x
 If u = F(x), x = F-1(u)
 We can generate u with a computer

Example: Exponential
 Example: Normal

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
80/108
Generating Random Samples

Exponential
u = F(x) = 1 – exp(-x)
 1 – u = exp(-x)
 x = (-1/ ) ln(1 – u)


Normal (,)
u = (z)
 z = -1(u)
 x = z +  =  -1(u) + 

600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
81/108
U[0,1] Generation
Linear congruential generator
x(n) = (a x(n-1) + b)mod m
 Properties of RNGs

Replicability – they are not RANDOM
 Period
 Randomness tests

The Mersenne twister: Current state of the
art (of pseudo-random number generation)
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Frequency
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
3 – Joint Distributions
Part 1 – Probability and Distribution Theory
83/108
Jointly Distributed Random
Variables
Usually some kind of association
between the variables. E.g., two
different financial assets
Joint cdf for two random variables
F(x, y) = Prob(X < x, Y < y)
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
84/108
Probability of a Rectangle
a1
b1
b2
Prob[a1 < x < b1, a2 < y < b2]
y
a2
x
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
100000
15000
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
300000
10
Mean
StDev
N
10
500000
400000
20
300000
200000
60
50
40
30
Normal
100
12
700000
600000
70
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
80
600000
200000
369687
156865
51
0.994
0.012
F(a1,a2)
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
+
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
F(a1,b2)
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
-
Frequency
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
F(b1,a2)
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
-
Percent
F(b1,b2)
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
85/108
Joint Distributions

Discrete: Multinomial for R kinds of
success in N independent trials

Continuous: Bi- and Multivariate
normal

Mixed: Conditional regression models
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
86/108
Multinomial Distribution
N trials, X1 ,...,X N
R different types of 'success'
Outcome: x1 ,...,x R such that

R
r 1
x r =N
 N  x1
xR
Prob[X1 =x1 ,...,X R =x R ] = 
p
...
p
 1
R
x
...
x
 1 R
Example: Mix of suits in 13 card bridge hand
13!
1
Prob[XS ,X D ,X C ,X H ] 
 
XS! X D! X C! X H !  4 
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
100000
15000
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
4
5
200000
2
1
100000
15000
0
200000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
300000
10
XH
Marginal Plot of Listing vs IncomePC
Mean
StDev
N
10
500000
400000
20
300000
200000
60
50
40
30
1
 
4
Normal
100
12
700000
600000
70
XC
Empirical CDF of Listing
14
800000
80
600000
200000
369687
156865
51
0.994
0.012
1
 
4
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
XD
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
1
 
4
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
XS
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
87/108
Probabilities: Inherited Color Blindness
Inherited color blindness has different incidence rates in men and women.
Women usually carry the defective gene and men usually inherit it.
Pick an individual at random from the population.
B=1 = has inherited color blindness, B=0, not color blind
G=0 = MALE = gender, G=1, Female
Marginal: P(B=1)
= 2.75%
Conditional:P(B=1|G=0)
= 5.0%
(1 in 20 men)
P(B=1|G=1)
= 0.5%
(1 in 200 women)
Joint:
P(B=1 and G=0)
= 2.5%
P(B=1 and G=1)
= 0.25%
Meatball
Garlic 5.0%
2.3%
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

Frequency

Listing

Percent

Listing

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
88/108
Marginal Distributions
Prob[X=x] = y Prob[X=x,Y=y]
Color Blind
Plain
32.5%
G=1
.4975
.0025
0.50
Total
.97255
.0275
1.00
600000
500000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Prob[G=0]=Prob[G=0,B=0]+
Prob[G=0,B=1]
Percent
Scatterplot of Listing vs IncomePC
900000
400000
Mushroom
16.2%
0.50
900000
Total
Frequency
Sausage
5.8%
.025
700000
Listing
Pepper and Onion
7.3%
.475
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
B=1
Listing
Pepperoni
21.8%
G=0
Listing
Meatball
Garlic 5.0%
2.3%
B=0
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Gender
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
89/108
Joint Continuous Distribution
Probability associated with a region A  
Prob((X,Y)  A)= 
A

f(x,y)dydx
f(x,y) is the joint density function
Joint CDF
For a region A = {(X,Y) with X  x, Y  y}
F(x,y) =
x
y


 
f(x,y)dydx
Joint density
 2 F(x,y)
f(x,y) =
xy
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
90/108
Marginal Distributions

If f(x,y) is the joint density, x  A, y  B, then f(y) =
A
f(x,y)dx
Example: f(x,y) = (x)e  ( x ) y , y  0, 0  x  1.

f(y) =
1
0
(λx)e-(λx)y dx. Make the change of variable to t = x
1  -ty
1  (-y)t
te
dt
=
te dt. Now let a = -y


0
0


1  at
  t 1  at  
f(y) =
te
dt
=
  - 2  e  0 where a = -y

0

 a a  
1
f(y) = 2 2 1-λye-λy -e-λy 
 y
f(y) =
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
91/108
Two Leading Applications

Copula Function - Application in Finance

Bivariate Normal Distribution
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
92/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
93/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
94/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
95/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
96/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
97/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
98/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
99/108
The Bivariate Normal Distribution
(x, y) distributed as bivariate normal if
(1) Infinite range - < x,y < +
(2) Normal marginal distributions:
2
2






y


1
1  x   x 
1

1
y
f(x) =
exp  
exp  
 
  , f(y) =

 2  y  
 x 2
 y 2
 2   x  


(3) Joint Density
2
2




y  y
 x  x
1
1  x   x 
f(x,y)=
exp 


2


 

2 
2

2(1


)


2 x  y 1  
x
 
y
 x



(6) New parameter  is the correlation between x and y.
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
  y   y  
 
 


y
 
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
100/108
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
101/108
Independent Random Variables
800000
800000
400000
Mushroom
16.2%
Plain
32.5%
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
700000
700000
500000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
Percent
900000
0
1000000
60
800000
40
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
600000
= 2 F(x,y)/x y
= f(x) f(y)
Frequency
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepper and Onion
7.3%
f(x,y)
Listing
Pepperoni
21.8%

Listing
Meatball
Garlic 5.0%
2.3%
F(x, y) = Prob(X < x, Y < y)
= Prob(X < x) Prob(Y < y)
= FX(x) FY(y)
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
102/108
Independent Normals
(x, y) distributed as bivariate normal

 x    2  y  
1

1

y
x
f(x,y) =
exp 





 2(1  2 )   x    y
2 x  y 1  2


(2) Uncorrelated;  = 0
2
 
 y  y

x


1

1

x
f(x,y) =
exp  

 
 2   x    y
2 x  y
 
2

 x  x

2



 x

  y   y  
 
 
   y  




 1  y    2 
 1  x    2  1
1
y
x
=
exp  
exp  
 
 

 2  y  
 x 2
 2   x    y 2





2
In the bivariate normal case (not generally), correlation = 0 implies indepencence
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
103/108
Conditional Distributions
Conditional Probability: Prob(X  A | Y  B)
Prob(X  A and Y  B)
=
Prob(Y  B)
In the discrete case: Prob(X = x | Y = y)
Prob(X = x and Y = y)
Prob(Y = y)
=
Color Blind
.025
0.50
G=1 (F)
.4975
.0025
0.50
Total
.97255
.0275
1.00
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
Listing
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Percent
Pepperoni
21.8%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
.475
Listing
Meatball
Garlic 5.0%
2.3%
Total
G=0 (M)
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
B=1
(Yes)
Percent
B=0
(No)
Frequency
Gender
Prob(Not color blind given male)
Prob(B=0|G=0)
= Prob(B=0,G=0) / Prob(G=0)
=
.475 / .50
=
.950
Prob(B=1|G=0) = .025/.5 = .05
Prob(B=1|G=0)+Prob(B=0|G=0)=1
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
104/108
Conditional Distribution
Continuous Normal
f (y, x)
f (x)
(x, y) distributed as bivariate normal
f(y|x) =

 x    2  y    2
 y   y  


x



1

y
x
x
f(x,y) =
exp 
  2 
 
  
 
2 
2

2(1


)




2 x  y 1  
x
 
y

x

y


 

 1  x   2 
1
x
f(x) =
exp  
 
2

 x 2
 
x
 
2
 

  y  

y

[



(

/

)(x


)]
1
1 
1
1 

y
y
x
x
y|x



  
f (y | x) 
exp
exp  

2 
   y|x 2
 2   y|x
 y 2 1   2
 y 1  2



 

 
1
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Marginal Plot of Listing vs IncomePC
Mean
StDev
N
10
500000
300000
0




Normal
100
12
700000
400000
10
2
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%



20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
105/108
Bivariate Normal
Joint distribution is bivariate normal
 Marginal distributions are normal
 Conditional distributions are normal

y
 y|x   Y  
x
(x   x )
 the conditional mean function
= the regression function of y on x.
Note :  y|x    x
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
106/108
Y and
Y|X
Y
X
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
X
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
107/108
Model Building
Pepperoni
21.8%
Sausage
5.8%
900000
800000
800000
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
900000
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
500000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
600000
300000
100000
Probability Plot of Listing
99
700000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Meatball
Garlic 5.0%
2.3%
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
Percent

Frequency

Listing

Typically f(y|x) is of interest
x is generated by a separate process f(x)
Joint distribution is f(y,x)=f(y|x)f(x)
Ex: demographic
y = log(household income|family size)
x = family size
 y|x ~ Normal(y|x , y|x )
 x ~ Poisson ()
Percent

20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000
Part 1 – Probability and Distribution Theory
108/108
X=4
X=3
X=2
X=1
y|x ~ Normal[ 20 + 3x, 42 ], x = 1,2,3,4; Poisson
600000
500000
400000
Mushroom
16.2%
Plain
32.5%
Scatterplot of Listing vs IncomePC
Normal - 95% CI
90
500000
400000
200000
100000
15000
60
50
40
30
17500
20000
22500
25000
IncomePC
27500
30000
32500
6
5
200000
2
1
100000
15000
400000
600000
Listing
800000
1000000
369687
156865
51
80
8
4
200000
Mean
StDev
N
10
500000
300000
0
Normal
100
12
700000
400000
10
Marginal Plot of Listing vs IncomePC
Empirical CDF of Listing
14
800000
600000
70
20
300000
200000
369687
156865
51
0.994
0.012
80
600000
Histogram of Listing
900000
Mean
StDev
N
AD
P-Value
95
700000
300000
100000
Probability Plot of Listing
99
17500
20000
22500
25000
IncomePC
27500
30000
32500
0
1000000
60
800000
40
Listing
800000
800000
Percent
900000
Frequency
Sausage
5.8%
Scatterplot of Listing vs IncomePC
900000
700000
Listing
Pepper and Onion
7.3%
Boxplot of Listing
Category
Pepperoni
Plain
Mushroom
Sausage
Pepper and Onion
Mushroom and Onion
Garlic
Meatball
Listing
Pepperoni
21.8%
Listing
Meatball
Garlic 5.0%
2.3%
Percent
Pie Chart of Percent vs Type
Mushroom and Onion
9.2%
20
600000
400000
0
0
200000
300000
400000
500000 600000
Listing
700000
800000
900000
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
10
20
30
40
50
60
70
80
90
Listing
200000
15000
20000
25000
IncomePC
30000