8 Joint Distributions.pptVIP

8. Joint Distributions 8.1 Bivariate distribution Def Let X and Y be 2 discrete random variables defined on the same sample space. Let the sets of possible values of X and Y be A and B, respectively. The function p(x, y)=P(X=x, Y=y) is called the joint probability mass function of X and Y. Bivariate distribution Def Let X and Y have joint probability mass function p(x,y). Let the sets of possible values of X and Y be A and B, respectively. Then the functions are called, respectively, the marginal probability mass functions of X and Y. Bivariate distribution Ex 8.1 A small college has 90 male and 30 female professors. An ad hoc committee of 5 is selected at random to write the vision and mission of the college. Let X and Y be the number of men and women on this committee, respectively. (a) Find the joint probability mass function of X and Y. (b) Find pX and pY, the marginal functions of X and Y. Bivariate distribution Sol: Bivariate distribution Ex 8.2 Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. What is joint probability mass function? and what are pX and pY? Bivariate distribution Sol: y pX(x) 0 1 2 3 4 5 6 x 1 1/12 1/12 0 0 0 0 0 1/6 2 1/24 2/24 1/24 0 0 0 0 1/6 3 1/48 3/48 3/48 1/48 0 0 0 1/6 4 1/96 4/96 6/96 4/96 1/96 0 0 1/6 5 1/192 5/192 10/192 10/192 5/192 1/192 0 1/6 6 1/384 6/384 15/384 20/384 15/384 6/384 1/384 1/6 pY(y) 63/384 120/384 99/384 64/384 29/384 8/384 1/384 Bivariate distribution Thm 8.1 Let f(x,y) be the probability mass function of discrete r. v. X and Y. If h is a function of two variables from R2 to R, then Z=h(X,Y) is a random variable with the expected value given by Corollary of Thm 8.1 E(X+Y)= E(X)+E(Y) Proof: Bivariate distribution Def Two r. v.’s X and Y, defined on the same sample space, have a continuous joint distribution if there exists a n