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Let X1, X2, . . . be independent random variables, with Xi = Bernoulli(pi) for some sequence of constants p1, p2, . . . . So, for example, P(X3 = 1) = p3, and P(X1 = 0 and X2 = 1) = (1 â p1)p2. i) For any k > 0, give P(Xi = 0 for every i â {1, . . . , k}). ii) Let N = min{n : Xn = 1}, and find P(N > n) for any n. iii) Find values of p1, p2, p3 so that P(N > 3) = 0.

Let X1, X2, . . . be independent random variables, with Xi = Bernoulli(pi) for some sequence of constants p1, p2,

. . . . So, for example, P(X3 = 1) = p3, and P(X1 = 0 and X2 = 1) = (1 â p1)p2.

i) For any k > 0, give P(Xi = 0 for every i â {1, . . . , k}).

ii) Let N = min{n : Xn = 1}, and find P(N > n) for any n.

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iii) Find values of p1, p2, p3 so that P(N > 3) = 0.

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"FIRST15"

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