Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n,...

Let X1, X2, X3, . . . be independently random variables such that Xn ∼ Bin(n, 0.5) for n ≥ 1. Let N ∼ Geo(0.5) and assume it is independent of X1, X2, . . .. Further define T = X_N .

(a) Find E(T) and argue that T is short proper.

(b) Find the pgf of T.

(c) Use the pgf of T in (b) to find P(T = n) for n ≥ 0.

(d) Use the pgf of T in (b) to find V ar(T).

Expert Solution

orchestra answered 1 year ago

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