5.4 Joint Distributions and
Independence
A joint probability function for discrete
random variables X and Y is a nonnegative
function f(x,y), giving the probability that
(simultaneously) X takes the value x and Y
takes the value y. That is
f(x,y)  P(X  x and Y  y)
The function f(x,y) is a joint probability
distribution or probability mass function of
the discrete random variable X and Y if
1. f(x, y)  0
2.
 f(x, y)  1
x
y
3. P(X=x, Y=y)=f(x,y)
Example
A large insurance agency services a number of
customers who have purchased both a homeowner’s
policy and an automobile policy from the agency.
For each type of policy, a deductible amount must be
specified. For an automobile policy, the choices are
$100 and $250, whereas for a homeowner’s policy, the
choices are 0, $100, and $200.
Suppose an individual with both types of policy is
selected at random from the agency’s files.
Let X=deductible amount on the auto policy and Y=the
deductible amount on the homeowner’s policy.
The joint probability table is
y
p(x,y)
x
100
250
0
0.2
0.05
100
0.1
0.15
200
0.2
0.3
Then p(100, 100)=P(X=100 and Y=100)
=P($100 deductible on both policies)=.1
y
p(x,y)
x
100
250
0
0.2
0.05
100
0.1
0.15
200
0.2
0.3
The probability P(Y≥100) is computed by
summing probabilities of all (x,y) pairs for
which y ≥100.
P(Y≥100)=p(100,100)+p(250,100)+p(100,200)
+p(250,100)=.75
From Joint probability to individual
distributions marginal distributions
• The joint probability function, f(x,y), of X and Y,
contains more information than individual
probabilities functions of X, and Y.
• Individual probabilities functions of X, and Y can be
obtained from their joint probability function.
• We call the individual probability functions of X and Y
marginal distributions.
Marginal Distributions
• Definition: The individual probability functions for
discrete random variables X and Y with joint
probability function f(x,y) are called marginal
probability functions. They are obtained by summing
f(x,y) values over all possible values of the other
variable.
• The marginal probability function for X is
f X ( x )   f ( x, y )
y
• And the marginal probability function for Y is
f Y ( y )   f ( x, y )
x
Marginal probabilities
y
p(x,y)
x
fY(y)
100
250
0
0.2
0.05
0.25
100
0.1
0.15
0.25
200
0.2
0.3
0.5
fX(x)
0.5
0.5
Conditional Distributions
• For discrete random variables X and Y with joint
probability function f(x,y), the conditional
probability function of Y given X=x is
f ( x, y )
fY | X ( y | x ) 
f X ( x)
• Similarly, the conditional distribution of X given Y=y
is
f X |Y ( x | y ) 
f ( x, y )
fY ( y )
Example
• Given the joint probabilities and marginal
probabilities
f(x,y)
x= 0
x=1
x=2
fY(y)
y=0
1/6
2/9
1/36
15/36
y=1
1 /3
1/6
0
1/2
y=2
1/12
0
0
1/12
fX(x)
7/12
7/18
1/36
1
• Find the probability of Y , given X=0.
Solution
f (0, y )
fY | X ( y | 0) 
f X (0)
• fX(0)=7/12
• f(0,0)=1/6, f(0,1)=1/3, f(0,2)=1/12
Then
• fY|X(0|0)=2/7; fY|X (1|0)=4/7, fY|X (2|0)=1/7
Statistical Independence
f(x|y) doesn’t depend on y;
f(y|x) doesn’t depend on x.
• Verify that f(x|y)=fX(x) & f(y|x)= fY(y).
f(x,y)=f(x|y) fY(y)
f(x,y)= fX(x) fY(y).
• Then X and Y are independent
Definition
• Discrete random variables X and Y are called
independent if their joint probability function
f(x,y) is the product of their respective marginal
distributions.
• That is, X and Y are said to be statistically
independent if
f(x,y)=fX(x)fY(y)
for all x,y.
Example
• X and Y have the following joint distribution
function
F(x,y)
x=1
2
3
y=1
0.16
0.16
0.08
2
0.24
0.24
0.12
• Verify that X and Y are independent.
Example
• X and Y have the following joint distribution
function
F(x,y)
x=1
2
3
y=1
0.16
0.08
0.16
2
0.24
0.24
0.12
• Verify that X and Y are dependent.