Let X and Y be two independent random variables with X ~ Poisson(µy). Poisson(ux) and Y (a) Show that the moment generating function (mgf) of X is Mx(t) = exp{µx(e' – 1)} = eHx(et –1). (b) Show that the mgf of X + Y is Mx+y(t) = exp{(ux + µy)(e' – 1)} (c) Based on the result in (b), what distribution does X +Y follow? (d) Use the mgf you found in (a) to show that E(X²) = µ + µx.

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