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Let $ X_1,...,X_n $ be independent geometric distributed random variables with parameter $p$.

($a$): Show that $S_n = X_1 + ...+ X_n $ is negative binomial distributed with parameter $n$ and $p$.

($b$): Find the conditional distribution of $(X_1,...,X_n)$ given $S_n = s $.

($c$): Find the conditional distribution of $X_1$ given $S_n = s $.

($d$): Consider $n \in $ {2,3} . Find $g(s) := \mathbb{E}(X_1| S_n =s)$ and calculate $\mathbb{E}(g(S_n))$.

Attempt:

a) clear to me. ( induction ) So I don't need help here.

b) Do I have to consider $P(X_1 = x_1,...,X_n=x_n | S_n=s) $ ? If this is correct I would have: $P(X_1 = x_1,...,X_n=x_n | S_n=s) = \frac{P(X_1 = x_1,...,X_n=x_n)}{P(S_n=s)} $ And this is $\frac{1}{\binom{n+s-1}{s}}$, I think.

c) clear to me. Result: $ \frac{(n+s-k-2)!s!(n-1)}{(n+s-1)!(s-k)!} $. Do you see a way to simplify this?

d) I'm stucked here. I know $\mathbb{E}(Y| X= k) := \sum_y yP(Y=n|X=k) $ and that we have to use the law of total expectation for $\mathbb{E}(g(S_n))$.

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