5.4 Joint Distributions and Independence

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 Then p(100, 100)=P(X=100 and Y=100) =P($100 deductible on both policies)=.1

6. 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 probabilities

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. 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  1/6  0  1/2  1/12  0  0  1/12 fX(x)  7/12  7/18  1/36 1 Example • Given the joint probabilities and marginal probabilities • Find the probability of Y , given X=0.

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.

16. Example • X and Y have the following joint distribution function • Verify that X and Y are dependent.