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Question: Suppose that N is a Poisson random variable with parameter µ. Suppose that given N = n, random variables X1, X2, . .. Xn are independent with uniform (0, 1) distribution so there are a random number of X's.

a) Given N = n, what is the probability that all the X's are less than t?

b) What is the (unconditional) probability that all the X's are less than t?

c) Let SN = X1 + . . . +XN denote the sum of the random number of X's, (If N = 0 then SN = 0.) Find P(SN = 0). Explain.

d) Find E(SN).

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