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If we think of W 1 as the number of trials we have to make to get the first success, and then W 2 the number of further trials to the second success, and so on, we can see that X = W 1 W 2 W r, and that the W i are independent and geometric random variables So EX = r/p, and Var(X) = r(1−p)/p2 5 Poisson random variablesP R I O R I T Y D a t e 0 2 / 2 1 / 2 0 0 74 1222 (d) Prove that f(f−1(B)) = B for all B ⊆ Y iff f is surjective Proof =⇒ Let y ∈ Y arbitrary We have to show that there exists x ∈ X with f(x) = y Let B = {y} By assumption, f(f−1(B)) = B = {y}, so y ∈ f(f−1(B))By definition this means that there exists x ∈ f−1(B) with f(x) = y

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