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Chapter 3
Discrete Random Variables and Probability Distributions
 3.1 - Random Variables
 3.2 - Probability Distributions for Discrete
Random Variables
 3.3 - Expected Values
 3.4 - The Binomial Probability Distribution
 3.5 - Hypergeometric and Negative
Binomial Distributions
 3.6 - The Poisson Probability Distribution
What is the connection between
probability and random variables?
Events (and their corresponding
probabilities) that involve
experimental measurements can
be described by random variables.
2
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
x1
x2
x3
x6
…etc….
x5
x4
xn
SAMPLE of size n
Pop values
Probabilities
xi
p(xi )
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
Data values
Relative Frequencies
xi
p(xi ) = fi /n
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
xk
p(xk)
Total
1
3
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
“Density”
f ( x ) p ( x)
(height) (area)
Probability
Histogram
p( x)  f ( x) x
Probabilities
x
p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
Total Area = 1
p(x) = Probability that the
random variable X is equal
to a specific value x, i.e.,
|
x
x
(width)
Pop values
p(x) = P(X = x)
“probability mass
function” (pmf)
|
x
X
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 Histogram
Probability Table
x
p(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
1
Density f(x)
P(X = x)
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 (4)  P( X  4) 
5
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 Histogram
Probability Table
x
p(x)
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
1
Density f(x)
P(X = x)
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 (4)  P( X  4) 
1
6
6
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Probability
Histogram
Pop values
Probabilities
x
p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
Total Area = 1
F(x) = Probability that the
random variable X is less
than or equal to a specific
value x, i.e.,
F(x) = P(X  x)
“cumulative distribution
function” (cdf)
|
x
X
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
p(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
8
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
p(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
9
POPULATION
Pop vals
pmf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
random variable X
Example: X = Cholesterol level (mg/dL)
cdf
Calculating
“interval probabilities”…
F(b) = P(X  b)
F(a–) = P(X  a–)
F(b) – F(a–) =
P(X  b) – P(X  a–)
= P(a  X  b)
b
  p(x)
a
| |
a–a
|
b
X
F(x) = P(X  x)
POPULATION
Pop vals
pmf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
random variable X
Example: X = Cholesterol level (mg/dL)
Calculating
“interval probabilities”…

F(b) = P(X  b)
F(a–) = P(X  a–)
b
a
cdf
f ( x) dx  F (b)  F (a)
b

f
(
x
)

x

F
(
b
)

F
(
a
)

F(b) – F(a–) =
a
p( x)
P(X  b) – P(X  a–)
= P(a  X  b)
b
  p(x)
a
F(x) = P(X  x)
| |
a–a
|
b
X
FUNDAMENTAL
THEOREM OF
CALCULUS
(discrete form)
POPULATION
Pop vals
pmf
x
p(x)
x1
p(x1)
F(x1) = p(x1)
x2
p(x2)
F(x2) = p(x1) + p(x2)
x3
p(x3)
F(x3) = p(x1) + p(x2) + p(x3)
⋮
⋮
⋮
Total
1
increases from 0 to 1
random variable X
Example: X = Cholesterol level (mg/dL)
Calculating
“interval probabilities”…

F(b) = P(X  b)
F(a–) = P(X  a–)
b
a
cdf
f ( x) dx  F (b)  F (a)
b

f
(
x
)

x

F
(
b
)

F
(
a
)

F(b) – F(a–) =
a
p( x)
P(X  b) – P(X  a–)
= P(a  X  b)
b
  p(x)
a
F(x) = P(X  x)
| |
a–a
|
b
X
FUNDAMENTAL
THEOREM OF
CALCULUS
(discrete form)
Chapter 3
Discrete Random Variables and Probability Distributions
 3.1 - Random Variables
 3.2 - Probability Distributions for Discrete
Random Variables
 3.3 - Expected Values
 3.4 - The Binomial Probability Distribution
 3.5 - Hypergeometric and Negative
Binomial Distributions
 3.6 - The Poisson Probability Distribution
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
random variable X
Example: X = Cholesterol level (mg/dL)
Just as the sample mean x and sample variance s2 were used to characterize
“measure of center” and “measure of spread” of a dataset, we can now define the
“true” population mean  and population variance  2, using probabilities.
•
Population mean
   x p ( x)
Also denoted by E[X], the “expected value” of the variable X.
•
Population variance
 2   ( x   ) 2 p ( x)
14
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
random variable X
Example: X = Cholesterol level (mg/dL)
Just as the sample mean x and sample variance s2 were used to characterize
“measure of center” and “measure of spread” of a dataset, we can now define the
“true” population mean  and population variance  2, using probabilities.
•
Population mean
   x p ( x)
Also denoted by E[X], the “expected value” of the variable X.
•
Population variance
 2   ( x   ) 2 p ( x)
15
Example 1: POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
1/2
Pop values
Probabilities
xi
p(xi )
210
1/6
240
1/3
270
1/2
Total
1
1/3
1/6
   x p( x)  (210)(1/ 6)  (240)(1/ 3)  (270)(1/ 2)  250
2
2
2
 2   ( x   )2 p( x)  (40) (1/ 6)  (10) (1/ 3)  (20) (1/ 2)  500
16
Example 2: POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Equally likely outcomes result
in a “uniform distribution.”
Pop values
Probabilities
xi
p(xi )
180
1/3
210
1/3
240
1/3
Total
1
1/3
1/3
1/3
   x p( x)  (180)(1/ 3)  (210)(1/ 3)  (240)(1/ 3)  210 (clear from symmetry)
2
2
2
 2   ( x   )2 p( x)  (30) (1/ 3)  (0) (1/ 3)  (30) (1/ 3)  600
17
To summarize…
18
POPULATION
Discrete
random variable X
Probability Table
Pop
Probabilities
xi
pmf p(xi )
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
1
Probability Histogram
Total Area = 1
X
   x p( x)
 2   ( x   ) 2 p ( x)
Frequency Table
Data
xi
x1
x2
x3
x6
x4
…etc….
x5
xn
SAMPLE of size n
Relative
Frequencies
Density Histogram
p(xi ) = fi /n
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
xk
p(xk)
1
Total Area = 1
X
x   x p( x)
s 2  nn1  ( x  x ) 2 p( x)
19
POPULATION
Continuous
Discrete
random variable X
Probability Table
Pop
Probabilities
xi
pmf p(xi )
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
1
Probability Histogram
Total Area = 1
X
   x p( x)
 2   ( x   ) 2 p ( x)
Frequency Table
Data
xi
x1
x2
x3
x6
x4
…etc….
x5
xn
SAMPLE of size n
Relative
Frequencies
Density Histogram
p(xi ) = fi /n
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
xk
p(xk)
1
Total Area = 1
X
x   x p( x)
s 2  nn1  ( x  x ) 2 p( x)
20
One final example…
21
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
D = X1 – X2 ~ ???
d
-30
0
Outcomes
(210, 240)
(210, 210), (240, 240)
+30
(210, 180), (240, 210), (270, 240)
+60
(240, 180), (270, 210)
+90
(270, 180)
22
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
D = X1 – X2 ~ ???
d
-30
0
Probabilities
Outcomesp(d)
1/9
? 240)
(210,
2/9
? 210), (240, 240)
(210,
+30
3/9
? 180), (240, 210), (270, 240)
(210,
+60
2/9
? 180), (270, 210)
(240,
+90
1/9
? 180)
(270,
The
outcomes of
D are NOT
EQUALLY
LIKELY!!!
23
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
D = X1 – X2 ~ ???
d
-30
0
Probabilities
Outcomesp(d)
(1/6)(1/3)
(210, 240)= 1/18 via independence
(210, 210), (240, 240)
+30
(210, 180), (240, 210), (270, 240)
+60
(240, 180), (270, 210)
+90
(270, 180)
24
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
D = X1 – X2 ~ ???
d
-30
0
Probabilities p(d)
(1/6)(1/3) = 1/18 via independence
(210, 210),+ (1/3)(1/3)
(1/6)(1/3)
(240, 240)
= 3/18
+30
(210, 180), (240, 210), (270, 240)
+60
(240, 180), (270, 210)
+90
(270, 180)
25
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
Probability Histogram
6/18
5/18
3/18
3/18
1/18
D = X1 – X2 ~ ???
d
-30
0
Probabilities p(d)
(1/6)(1/3) = 1/18 via independence
(1/6)(1/3) + (1/3)(1/3) = 3/18
+30
(210, 180),+ (1/3)(1/3)
(240, 210),
(270, 240)
(1/6)(1/3)
+ (1/2)(1/3)
= 6/18
+60
(240, 180),+ (1/2)(1/3)
(270, 210)
(1/3)(1/3)
= 5/18
+90
(270, 180)= 3/18
(1/2)(1/3)
26
Example 3: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
Probability Histogram
X2 = Cholesterol level (mg/dL)
x
p1(x)
1 = 250
x
p2(x)
2 = 210
210
1/6
12 = 500
180
1/3
22 = 600
240
1/3
210
1/3
270
1/2
240
1/3
Total
1
Total
1
D = X1 – X2 ~ ???
d
-30
0
6/18
5/18
3/18
1/18
D = (-30)(1/18) + (0)(3/18) +
(30)(6/18) + (60)(5/18) +
(90)(3/18) = 40
Probabilities f(d)
D = 1 – 2
(1/6)(1/3) = 1/18 via independence
(1/6)(1/3) + (1/3)(1/3) = 3/18
+30
(210, 180),+ (1/3)(1/3)
(240, 210),
(270, 240)
(1/6)(1/3)
+ (1/2)(1/3)
= 6/18
+60
(240, 180),+ (1/2)(1/3)
(270, 210)
(1/3)(1/3)
= 5/18
+90
(270, 180)= 3/18
(1/2)(1/3)
3/18
D2 = (-70) 2(1/18) + (-40) 2(3/18) +

(-10) 2(6/18) + (20) 2(5/18) +
(50) 2(3/18) = 1100
2 =
2 +
2
D
1
2


27
General: TWO INDEPENDENT POPULATIONS
X1 = Cholesterol level (mg/dL)
IF the two
Probability
Histogram
populations
are
dependent…
X2 = Cholesterol level (mg/dL)
x
f1(x)
1 = 250
210
1/6
12 = 500
240
1/3
f2(x) 2 = 210
…then
this
2
180
1/3still 
formula
holds,
2 = 600
210 BUT……
1/3
270
1/2
240
Total
1
x
1/3
-30
0
5/18
3/18
3/18
1/18
Mean (X1 – X
Total
2) = 1Mean (X1) – Mean (X2)
D = X1 – X2 ~ ???
d
6/18
D = (-30)(1/18) + (0)(3/18) +
(30)(6/18) + (60)(5/18) +
(90)(3/18) = 40
Probabilities f(d)
D = 1 – 2
(1/6)(1/3) = 1/18 via independence
(1/6)(1/3) + (1/3)(1/3) = 3/18
= (-70)
+ Cov
(-40) 2(3/18)
+ )
Var (X1 – X2) = Var (X1) D+2 Var
(X22(1/18)
)
–
2
(X
,
X
2
1
2
2
+30
(210, 180),+ (1/3)(1/3)
(240, 210),
(270, 240)
(1/6)(1/3)
+ (1/2)(1/3)
= 6/18
+60
(240, 180),+ (1/2)(1/3)
(270, 210)
(1/3)(1/3)
= 5/18
These two formulas are valid for
(270, 180)
+90
(1/2)(1/3)
= 3/18
continuous
as well
as discrete distributions.

(-10) (6/18) + (20) (5/18) +
(50) 2(3/18) = 1100
2 =
2 +
2
D
1
2


28
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Suppose X is transformed to another random variable, say h(X).
Then by def, h ( X )  E[h( X )] 
 h( x) p( x)
Variance:  X2  
E ( xXXX))22 p(x
) ( x   X ) 2 p( x)
29
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
bx1
bx2
bx3
p(x1)
⋮
⋮
Total
1
p(x2)
p(x3)
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Suppose X is constant, say b, throughout entire population…
Then by def,
E[b] 
 b p ( x) 
b
 p ( x)
 b 1  b
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
30
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
bx1
bx2
bx3
p(x1)
⋮
⋮
Total
1
p(x2)
p(x3)
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Suppose X is constant, say b, throughout entire population…
Then…
E[b]  b
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
31
POPULATION
random variable X
Pop values
Probabilities
x
pmf p(x)
a x1
a x2
a x3
Example: X = Cholesterol level (mg/dL)
p(x1)
p(x2)
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Multiply X by any constant a…
Then by def,
E[aX ]   a x p( x)  a
 x p ( x)
 a E[ X ]
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
32
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
a x1
a x2
a x3
p(x1)
p(x2)
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Multiply X by any constant a…
Then…
E[aX ]  a E[ X ]
i.e.,…
a X  a  X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
33
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1  b
random variable X
Example: X = Cholesterol level (mg/dL)
x2  b
x3  b
p(x1)
p(x2)
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Multiply X by any constant a…
Then…
E[aX ]  a E[ X ]
i.e.,…
a X  a  X
Add any constant b to X…
 ( x  b) p( x)
  x p( x)   b p( x)
E[ X  b] 
 E[ X ]  E[b]
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
34
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1  b
random variable X
Example: X = Cholesterol level (mg/dL)
x2  b
x3  b
p(x1)
p(x2)
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
Multiply X by any constant a…
Add any constant b to X…
Then…
E[aX ]  a E[ X ]
E[ X  b]  E[ X ]  b
i.e.,…
a X  a  X
X b  X  b
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
35
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
random variable X
Example: X = Cholesterol level (mg/dL)
General Properties of “Expectation” of X
Mean:  X  E[ X ]   x p( x)
E[aX  b]  a E[ X ]  b
 a X b  a  X  b
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
36
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
Multiply X by any constant a… then X is also multiplied by a.
2
 aX
 E (aX  a X ) 2 
 E  a 2 ( X   X ) 2 
 a 2 E ( X   X ) 2 
 a 2  X2
37
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
Multiply X by any constant a… then X is also multiplied by a.
2
 aX
 a 2  X2
2
i.e.,…Var (aX )  a Var ( X )
 aX  a  X
i.e.,…SD(aX )  a SD( X )
38
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
Add any constant b to X…
then b is also added to X .
2
 X2 b  E  ( X  b)  ( X  b)  


 E  ( X   X ) 2 
  X2
39
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
Add any constant b to X…
then b is also added to X .
 X2 b   X2
i.e.,…Var ( X  b)  Var ( X )
 X b   X
i.e.,… SD( X  b)  SD( X )
40
POPULATION
Pop values
Probabilities
x
pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
random variable X
Example: X = Cholesterol level (mg/dL)
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
 E  X 2  2 X  X   X 2 
 E  X 2   2E 2X E
X XX   
EX2EX21
 E  X 2   2 X 2   X 2
 E  X 2    X 2
41
POPULATION
random variable X
Example: X = Cholesterol level (mg/dL)
Pop values
Probabilities
x
pmf pmf p(x)
x1
p(x1)
x2
p(x2)
x3
p(x3)
⋮
⋮
Total
1
General Properties of “Expectation” of X
Variance:  X2  E ( X   X ) 2    ( x   X ) 2 p( x)
 X2  E  X 2    X 2   x2 p( x)   X 2
  E  X   E[ X ]
2
X
2
2
  x p( x)   x p( x) 
2
2
This is the analogue of the “alternate computational
formula” for the sample variance s2.
42