1. 5.4 Joint Distributions and Independence

2. 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

3. The function f(x,y) is a joint probability distribution or probability mass function of the discrete random variable X and Y if 1. 2.
3.  P(X=x, Y=y)=f(x,y)

4. 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.

5. The joint probability table is
p(x,y)
100
200 x
0.2
0.1
250
0.05
0.15
0.3
Then p(100, 100)=P(X=100 and Y=100)
=P($100 deductible on both policies)=.1

6. y
p(x,y)
100
200 x
0.2
0.1
250
0.05
0.15
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

7. 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.

8. 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
And the marginal probability function for Y is

9. Marginal probabilitiesy
p(x,y)
100
200
fX(x) x
0.2
0.1
0.5
250
0.05
0.15
0.3
fY(y)
0.25

10. 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
Similarly, the conditional distribution of X given Y=y is

11. Example Given the joint probabilities and marginal probabilities
Find the probability of Y , given X=0.
f(x,y)
x= 0
x=1
x=2
fY(y)
 y=0
 1/6
 2/9
 1/36
 15/36
y=1 
y=2
 1 /3  0
 1/2
 1/12
 fX(x)
 7/12
 7/18 1

12. Solution 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

13. 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

14. 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.

15. Example X and Y have the following joint distribution function
Verify that X and Y are independent.
F(x,y)
x=1 2 3
y=1
0.16
0.08
0.24
0.12

16. Example X and Y have the following joint distribution function
Verify that X and Y are dependent.
F(x,y)
x=1 2 3
y=1
0.16
0.08
0.24
0.12