Where have we been?
Discrete Random Variables
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STAT 200
Tom Ilvento
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Moving towards
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We want to shift towards inferential
properties of statistics
To do this we need some sense of
probability distributions for discrete and
continuous random variables
Random Variables
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Expected Values and Variance of
Random Variables
Special Distributions
n Binomial Distribution
n Normal Distribution
Central Limit Theorem
And into inferences
n Single sample
n Two samples
Random Variables – variables that
assume numerical values associated
with random outcomes from an
experiment
n Discrete
n Continuous
Probability Distributions for Random
Variables
Example of a random
variable
This will lead us to
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We started with an introduction to
statistics
and the descriptive nature of statistics
for qualitative and quantitative variables
And the basic ideas and tools of
probability
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Suppose we were recording the
number of dentists that use nitrous
oxide (laughing gas) in their
practice
We know that 60% of dentists use
the gas.
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Nitrous Oxide Example
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Nitrous Oxide Example
Let X = number of dentists in a random
sample of five dentists that use use
laughing gas.
X is a random variable that can take on
the following values:
n 0, 1, 2, 3, 4, 5
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X
Nitrous Oxide Example
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0
P(X)
.0102
1
2
3
0
1
2
3
4
5
Nitrous Oxide Example
And then assign probabilities to each
value of the random variable X
X
We can list the values of our random
variable X
4
5
.0768 .2304 .3456 .2592 .0778
X
0
1
P(X)
.0102
2
3
4
5
.0768 .2304 .3456 .2592 .0778
What is the probability of 4 of 5 dentists
selected randomly Using laughing gas?
Note: we will look at how we assign
probabilities to this random variable later.
Nitrous Oxide Example
Nitrous Oxide Example
Probability
Distribution of X
What is the probability of less than 2 of 5
using laughing gas?
0.4
p(X)
0.3
If I randomly selected 5 dentists, how many
would I expect to use laughing gas?
0.2
0.1
0
0
1
2
3
4
5
Number of Dentists
2
How can you tell it is a
discrete random variable?
Types of Random Variables
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The variable in the dentist example is
called a discrete random variable
n Finite number of distinct possible
values
n We can assume that the values can
be listed or counted
Random Variables that fall along points
on an interval, and can’t be fully
counted, are call Continuous Random
Variables
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To describe a discrete
random variable
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Countable
Usually is described as “the number
of…”
Tends to be whole numbers
n Number of students applying to a
university
n Number of errors on a test
n Number of bacteria per cubic
centimeter of water
n Number of heart beats of a patient
Specifying a discrete random
variable – Tossing two coins
Specify the possible values it can
assume
Assign probabilities to each value
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Let X=number of heads observed
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So X takes on the following values
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Specifying a discrete random
variable – Tossing two coins
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We can a priori assign probabilities to X
Number of
Heads
Sample
Points
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TT
.25
1
HT TH
.50
2
HH
.25
O
1
2
TT
HT or TH
HH
Specifying a discrete random
variable – Tossing two coins
p(x)
Probabilities
0
Heads Heads
Heads Tails
Tails Heads
Tails Tails
This completely
defines the discrete
random variable X
Connecting
probabilities to the
values results in the
probability
distribution
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
The probabilities are distributed
over the values
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Probability Distribution: p(x)
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Can be shown by a
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Graph
Table
Formula – will come later
Probability Distribution: p(x)
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Specifies the probability associated with
each value
Requirements:
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p(x) $0
3p(x) = 1
for all values of x
Probability Distributions of
Discrete Random Variables
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Sometimes the probabilities are known a
priori
Sometimes they are observed in
experiments, a posteriori
And sometimes we use a formula which
seems to apply to a class of discrete
random variables, such as the Binomial
Distribution
Probability of one male in
three births
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Notation
n p(x=0) = .25
n p(x=1) = .50
n p(x=2) = .25
Here’s our approach
n Show the sample points in three
births by showing all the possible
outcomes in three births
n Let X= number of male offspring
n Assign probabilities and depict the
probability distribution
n Sum the probabilities for at least one
male
Example: Problem 4.19 page 172
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Males have XY chromosomes
Females have XX chromosomes
Assume each are equally likely to be
contributed
If a couple has three children, what
is the probability of at least one
male?
Probability of one male in
three births
Birth 1
Birth 2
Birth 3
F
M
M
F
M
F
2x2x2=8
F
F
M
Combinations
of males and
females in
three births
F
F
F
M
M
M
M
M
F
M
F
M
M
F
F
There are:
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Probability of one male in
three births
Number of
Males
Sample
Points
p(X)
0
FFF
1/8
1
FMF FFM
MFF
3/8
2
FMM MMF
MFM
3/8
3
MMM
1/8
Probability of one male in
three births
Probability
Distribution p(X)
Let X=
number of
male
offspring and
find it’s
probability
distribution
0.4
0.3
0.2
0.1
0
0
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Probability of at least one male in three
births
= P(1 male) + P(2 males) + P(3 males)
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We could have solved this using the
compliment –
n 1 – P(no males) = 1 – 1/8 = 7/8
Mean and Variance of a
Discrete Variable
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We can think of our observed probability
distribution for x as having a mean and
variance
That are identical to the population
mean and variance
For the population
n : represents the mean
n F 2 represents the variance
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The relationship between observed
relative frequency distribution and the
population
Probability of one male in
three births
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If we were to observe a large number of
observations of a discrete random
variable x
The relative frequency distribution of x
Would begin to resemble the
probability distribution for the
population
Expected Values of Discrete
Random Values
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An expected value is another term for the
mean when dealing with a probability
distribution
The expected value of a discrete random
variable is
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E ( x ) = ∑ xi ⋅ p ( xi ) = µ
i =1
The sum of each value times the
probability of that value
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The Expectation of a Discrete
Variable
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I take each value of the discrete
variable
Multiply by the probability associated
with the value
Sum the calculations
Expectation of the Male Baby
Example
Number
of Males
p(X)
0
FFF
1/8 = .125
1
FMF FFM
MFF
3/8 = .375
2
FMM MMF
MFM
3/8 = .375
3
MMM
1/8 = .125
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E ( x ) = ∑ xi ⋅ p ( xi ) = µ
Sample
Points
i =1
Expectation of the Male Baby
Example
The Variance of a Discrete
Variable
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The variance of a discrete random
variable is given as the
Expectation of the squared deviations
about the population mean:
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E[(x − µ )2 ] = ∑ ( xi − µ )2 p( xi ) = σ 2
i =1
Sum of the squared deviation of each value from the
mean times the probability of the value
The Variance of a Discrete
Variable
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I take each value of the discrete variable
Subtract the mean
Square the result
Multiply by the probability associated with
the value
Sum the calculations
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E[( x − µ ) 2 ] = ∑ ( xi − µ ) 2 p( xi ) = σ 2
i =1
Variance of the Male Baby
Example
Number
of Males
Sample
Points
p(X)
0
FFF
1/8 = .125
1
FMF FFM
MFF
3/8 = .375
2
FMM MMF
MFM
3/8 = .375
3
MMM
1/8 = .125
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Variance of the Male Baby
Example
Mean and Variance of Discrete
Random Variable
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Nitrous Oxide Example
X
0
1
P(X)
.0102
2
3
4
5
If I randomly selected 5 dentists, how many
would I expect to use laughing gas?
X
0
P(X)
.0102
• E(X) = 3
• S2 = 1.1998
• S = 1.09535
n
p(X)
0.2
n
0
0
1
2
3
4
2
3
4
5
.0768 .2304 .3456 .2592 .0778
So what?
0.3
0.1
1
If I randomly selected 5 dentists, what is the
variance (F 2)?
Probability Distribution of the
Discrete Variable X
0.4
In our example, F = .866
Nitrous Oxide Example
.0768 .2304 .3456 .2592 .0778
Probability
Distribution of X
Remember, we said the variance is the
mean squared deviation about the mean
The standard deviation is the square
root of the variance
Knowing the mean and standard
deviation of a probability distribution
can be used with
n Chebyshev’s Rule
n Empirical Rule
We can find probabilities under the
curve
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Number of Dentists
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Solitaire Example
Solitaire Example
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Some results of one class
n =219
There are 53 outcomes of this discrete
random variable
n Dollars won/lost - -$52 to +$208
n Number of cards 0 to 52
In reality, there are some values that
just don’t happen
Student solitaire example
Dollars
Cards
Freq
P(x)
-$52
0
7
.0320
-$47
1
11
.0502
-$42
2
26
.1187
-$37
3
24
.1096
-$32
4
23
.1050
Solitaire Probability Distribution
0.1400
Expected Cards 8.2694
Expected Return $41.35
Net Return
-$10.65
0.1200
0.1000
0.0800
0.0600
0.0400
0.0200
53
49
45
.0411
41
9
37
52
33
0.0000
$208
29
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It is very hard to a priori determine the
odds of winning, so I ask students to help
in an experiment to see.
You can contribute by playing ten games
n It is best to play a game, record the
result in dollars, exit the game, and then
start again
n Send your results to ilvento@udel.edu
p(X)
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Solitaire Example
25
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21
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Under GAME picks OPTIONS
Pick draw three
Pick Vegas Style
Pick Keep Score
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Vegas Style plays a game where it costs
you $52 to play one hand.
The draw three options turns the deck over
three cards at a time and let’s you go
through the deck three times.
You get $5 for each card you place on the
top (i.e., Ace, two, three... of a suit).
If you get all 52 cards on top you would
get back $208
(52 x 5) - 52
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5
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I have been asking my students to help in
an experiment playing solitaire on the
computer
You need to get on any computer with
solitaire and set the game
We have been using the Vegas style
solitaire game.
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Solitaire Example
# Cards
8
Ilvento Solitaire results
Solitaire Example
52
49
46
43
11.506
$57.53
$5.53
40
37
34
31
28
25
22
19
16
13
7
10
4
Expected Cards
Expected Return
Net Return
1
p(X)
Solitaire Probability Distribution
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
# Cards
9