CHAPTER 4
• 4.1 - Discrete Models
 General distributions
 Classical: Binomial, Poisson, etc.
• 4.2 - Continuous Models
 General distributions
 Classical: Normal, etc.
Motivation ~ Consider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6.
Probability Table
Probability Histogram
P(X = x)
x
f(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
f (4)  P( X  4) 
1
2
Motivation ~ Consider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6.
Probability Table
Probability Histogram
P(X = x)
x
f(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
f (4)  P( X  4) 
1
6
1
3
Motivation ~ Consider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6.
Cumulative
distribution
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
1
4
Motivation ~ Consider the following discrete random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
X is said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6.
Cumulative
distribution
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
1
“staircase graph”
from 0 to 1
5
POPULATION
random variable X
Example: X = “reaction time”
“Pain Threshold” Experiment:
Volunteers place one hand on metal
plate carrying low electrical current;
measure duration till hand withdrawn.
Time
Time
intervals
intervals
= 1.0
= 5.0
0.5
2.0
1.0
secs
secs
“In the limit…”
we obtain a
density curve
Total Area = 1
SAMPLE
In principle, as # individuals in samples
increase without bound, the class
interval widths can be made arbitrarily
small, i.e, the scale at which X is
measured can be made arbitrarily fine,
since it is continuous.
6
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f(x) = density function
• f(x)  0
• Area = 1
00 F(x) increases
continuously
from 0 to 1.
x
x
x
f(x) no longer represents the probability P(X = x), as it did for discrete variables X.
In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)
However, we can define “interval probabilities” of the form P(a  X  b), using F(x).
7
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
F(b)
f(x) = density function
F(b)  F(a)
F(a)
• f(x)  0
• Area = 1
F(x) increases
continuously
from 0 to 1.
a
b
a
b
f(x) no longer represents the probability P(X = x), as it did for discrete variables X.
In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…)
However, we can define “interval probabilities” of the form P(a  X  b), using F(x).
8
“In the limit…” we obtain a density curve
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
F(b)
f(x) = density function
F(b)  F(a)
F(a)
• f(x)  0
• Area = 1
F(x) increases
continuously
from 0 to 1.
a
b
a
b
f(x) no longer represents the probability P(X = x), as it did for discrete variables X.
An “interval probability” P(a  X  b) can be calculated as the amount of area under
the curve f(x) between a and b, or the difference P(X  b)  P(X  a), i.e., F(b)  F(a).
(Ordinarily, finding the area under a general curve requires calculus techniques…
9
unless the “curve” is a straight line, for instance. Examples to follow…)
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
“In the limit…” we obtain a density curve
f(x) = density function
• f(x)  0
 = 1
• Area


f ( x ) dx  1
F(x) increases
continuously
from 0 to 1.
Thus, in general, P(a  X  b) =

Moreover,     x f ( x ) dx

b
a
f ( x ) dx = F(b)  F(a).
Fundamental
Theorem of
Calculus

2
2
and .    ( x   ) f ( x ) dx.
10
Consider the following continuous random variable…
Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)”
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
1
“staircase graph”
from 0 to 1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
P( X  4)  1 6
11
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
1
“staircase graph”
from 0 to 1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
P( X  4)  1 6
12
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Probability Table Cumul Prob
Probability Histogram
P(X = x)
P(X  x)
x
f(x)
F(x)
1
1/6
1/6
2
1/6
2/6
3
1/6
3/6
4
1/6
4/6
5
1/6
5/6
6
1/6
1
1
“staircase graph”
from 0 to 1
Density
Total Area = 1
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of rolling a 4?”
P( X  4)  1 6
13
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20 > 0

Cumul Prob
P(X  x)
F(x)
Density
Total Area = 1
1 Check?
1
1
1
6 Base 6= 6 – 16= 5 6
Height = 0.2
1
6
1
6
5  0.2 = 1 
X
“What is the probability of
that
rolling
a random
a 4?” child is 4 years old?” doesn’t mean…..
P( X  4)
4.000000000......)
 16
A single value is one point out of an infinite
continuum of points on the real number line.
The probability that a continuous
random variable is exactly equal to
any single value is ZERO!
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumul Prob
P(X  x)
Density
Alternate way using
cumulative distribution
function (cdf)…
1
6
1
6
1
6
1
6
1
6
F(x)
1
6
X
“What is the probability of
rolling
a 4?” child is 4between
4 and
5 years
old?”
that
a random
years old?”
actually
means....
P(4
( XX4) 5) = (5 – 4)(0.2) = 0.2
NOTE: Since P(X = 5) = 0, no change for P(4  X  5), P(4 < X  5), or P(4 < X < 5).
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumul Prob
P(X  x)
Density
Alternate way using
cumulative distribution
function (cdf)…
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of
rolling
a 4?” child is under 5 years old?
that
a random
F (5)  P( X  5)  0.2 (5  1)  0.8
F(x)
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumul Prob
P(X  x)
Density
Alternate way using
cumulative distribution
function (cdf)…
1
6
1
6
1
6
1
6
1
6
1
6
X
“What is the probability of
rolling
a 4?” child is under 4 years old?
that
a random
F (4)  P( X  4)  0.2 (4  1)  0.6
F(x)
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumul Prob
P(X  x)
Density
Alternate way using
cumulative distribution
function (cdf)…
1
6
1
6
1
6
1
6
1
6
F(x)
1
6
X
“What is the probability of
rolling
a 4?” child is between 4 and 5 years old?”
that
a random
P(4  X  5)  P( X  5)  P( X  4)
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumul Prob
P(X  x)
Density
Alternate way using
cumulative distribution
function (cdf)…
1
6
1
6
1
6
1
6
1
6
F(x)
1
6
X
“What is the probability of
rolling
a 4?” child is between 4 and 5 years old?”
that
a random
P(4  X  5)  P( X  5)  P( X  4)
= F(5)  F(4) = 0.8 – 0.6 = 0.2
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
Cumul
= Area under density curve
up toProb
x
Density
f ( x)  0.20
P(X  x)
F(x)
For any x, the area
under the curve is
1
6
1
6
1F(x) =10.2 (x1– 1). 1
6
6
6
6
x
X
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
f ( x)  0.20
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
Density
F(x) = 0.2 (x – 1)
For any x, the area
under the curve is
1
6
1
6
1F(x) =10.2 (x1– 1). 1
6
6
6
6
F(x) increases
continuously
from 0 to 1.
(compare with
“staircase graph”
for discrete case)
x
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  0.20
F(x) = 0.2 (x – 1)
Density
F(5) = 0.8
1
6
1
6
1
6
1
6
1
6
1
6
0.2
F(4) = 0.6
“What is the probability that a child is between 4 and 5?”
P(4  X  5)  P( X  5)  P( X  4)
= F(5)  F(4) = 0.8 – 0.6 = 0.2
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Further suppose that X is uniformly distributed over the interval [1, 6].
Density
f ( x)  .08 ( x  1) > 0 
1
Base
Height
 1)  (0.4)
Area = (6
2
=1 
0.4
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
0.4
Density
f ( x)  .08 ( x  1)
1
f (4)
Base
Height
 1)  ???
.08
(4  1)
Area = (4
2
“What is the probability that a child is under 4 years old?”
P( X  4)
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Density
f ( x)  .08 ( x  1)
1
Base
 1)  0.24
Area = (4
2
= 0.36
h=?
0.36
0.4
Alternate method, without
having to use f(x):
Use proportions via
similar triangles.
h
0.4

4 1 6 1
h  0.24
“What is the probability that a child is under 4 years old?”
P( X  4)  0.36
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Density
f ( x)  .08 ( x  1)
0.64
0.36
“What is the probability that a child is under 4 years old?”
“What is the probability that a child is over 4 years old?”
P( X  4)  0.36
P( X  4)  1  0.36
 0.64
Consider the following continuous random variable…
Example: X = “Ages of children from 1 year old to 6 years old”
Cumulative probability F(x) = P(X  x)
= Area under density curve up to x
f ( x)  .08 ( x  1)
Exercise…
Density
F(x) = ????????
x
P( X  4)  F (4)
“What is the probability that a child is under 5 years old?” P ( X  5)  F (5)
“What is the probability that a child is between 4 and 5?” P(4  X  5) 
“What is the probability that a child is under 4 years old?”
Unfortunately, the cumulative area (i.e.,
probability) under most curves either…
 requires “integral calculus,” or
 is numerically approximated and tabulated.
IMPORTANT SPECIAL CASE: “Bell Curve”
28