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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 $X_1, X_2,\ldots,X_n$ are i.i.d. with PMF $p(x) = P(X_i = x)$. Let $I \sim \mathrm{DUnif}(1, 2,\ldots,n)$ be the label of a randomly chosen store, so $X_I$ is the number of customers at a randomly chosen store.

(a) For $i$ not equal to $j$, find P$(X_i = X_j)$ in terms of a sum involving the PMF $p(x)$.

(b) Find the joint PMF of $I$ and $X_I$ . Are they independent?

(c) Does $X_I$ , the number of customers for a random store, have the same marginal distribution as $X_1$, the number of customers for store $1$?

(d) Let $J \sim \mathrm{DUnif}(1, 2,\dots,n)$ also be the label of a randomly chosen store, with $I$ and $J$ independent. Find $P(X_I = X_J)$ in terms of a sum involving the PMF $p(x)$. How does $P(X_I = X_J)$ compare to $P(X_I = X_J)$ for fixed $i$, $j$ with $i$ not equal to $j$?

Is my reasoning correct?

I did the following:

a) $P(X_i = X_j)=P(X_i = X_j=x)$=$\binom {n}{x}p^xq^{n-x}$

b)$P(I=i,X_i=x)=P(X_i=x|I=i)P(I=i)=p\cdot\frac1n$, Yes because store was choosen randomly

c) $P(X_i=x)=P(X_1=x)=p$

d) $P(X_i=X_j=x)=P(X_i=x)P(X_j=x)=p_ip_j$

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