There are n stores in a shopping center, labeled from 1 to n. Let Xi be the number of

Chapter 7, Problem 28

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There are n stores in a shopping center, labeled from 1 to n. Let Xi be the number of customers who visit store i in a particular month, and suppose that X1, X2,...,Xn are i.i.d. with PMF p(x) = P(Xi = x). Let I DUnif(1, 2,...,n) be the label of a randomly chosen store, so XI is the number of customers at a randomly chosen store. (a) For i 6= j, find P(Xi = Xj ) in terms of a sum involving the PMF p(x). (b) Find the joint PMF of I and XI . Are they independent? (c) Does XI , the number of customers for a random store, have the same marginal distribution as X1, the number of customers for store 1? (d) Let J DUnif(1, 2,...,n) also be the label of a randomly chosen store, with I and J independent. Find P(XI = XJ ) in terms of a sum involving the PMF p(x). How does P(XI = XJ ) compare to P(Xi = Xj ) for fixed i, j with i 6= j?