3. Let X1, X2,..., be a sequence of independent and identically distributed random variables. Let N be Poisson distributed with mean u and is independent of the X,'s. Define N W = -ΣΧ. i=1 We define W = 0 if N = 0. (a) Suppose each X; is normally distributed with mean 0 and variance 1. Work out the moment generating function for W given N. [4 marks] (b) Show that the moment generating function of W is given by Mw (t) = exp(ue/2 - µ), te R. [5 marks] (c) Calculate the mean and variance of W. [5 marks] (d) Now consider Z = NX1. Find the mean and variance of Z. [6 marks]

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