Let N be the number of phone calls made by the customers of a phone company in a given hour. Suppose that N ∼ Poisson(β), where β > 0 is known. Let Xi be the length of the i'th phone call, for i = 1, 2, . . . ,N. We assume X_i's are independent of each other and also independent of N. We further assume Xi ∼ Exponential(λ), where λ > 0 is known. Let Y be the sum of the lengths of the phone calls, i.e.,

Find EY and Var(Y).

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N Y r=ΣΧ. i=1