C OMER ECE 600 Homework 7 1. Let X and Y be two independent binomial random variables with pmf X Y (1 )0 p ( ) p ( )0 otherwise k n k n p p k n k k k-²-≤ ≤ ³ ´ µ = = ¶· ¸ ³ ¹ Let Z = X + Y. Find the pmf of Z. Do you find anything special about the pmf of Z? Comment on it. 2. The hat-check staff has had a long day, and at the end of the party they decide to return people’s hats at random. Suppose that n people have their hats returned at random. You have previously shown that the expected number of people who get their own hat back is 1, irrespective of the total number of people. In this problem you are asked to find the variance of the number of people who get their own hat back. Let X i = 1 if person i gets his or her own hat back and 0 otherwise. Let 1 S X n n i i = = º be the total number of people who get their own hat back. a. Show that Var(S n ) = 1. b. Explain why you cannot use the variance of sums formula to calculate Var(S n ). 3.
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