Question: 4. (a) Show that the moment generating function of a Poisson random variable with parameter 1 at point t is ed(et-1). (5 pt) (b) Show

4. (a) Show that the moment generating function of a Poisson random variable with parameter 1 at point t is ed(et-1). (5 pt) (b) Show that if X and Y are two independent Poisson random variables with parameters 1 and 12 respectively, then X + Y ~ Poisson()1 + 12). (5 pt) (c) If Sn ~ Poisson(n), find the limiting distribution of Sn-n. (5 pt) (d) Suppose Xn ~ Binomial(n, A) for some > > 0. Show that Xn = Poisson (1). (5 pt)

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