RefertoExercise2.66and the momentgeneratingfunction m(t) = E‰etY Ž, analternativeto the probabilityfunctionforcharacterizingaprobabilitydistribution.Forindependentrandom variables Y1, Y2, …, Yn, let T = Y1 + ⋯+ Yn.

(a) Explainwhy T has mgf determined bytheseparateonesas m(t) = m1(t)m2(t)⋯mn(t).

(b) Ifeach Yi ∼ N(μ, σ2), usethe mgf m(t) = eμt+σ2t2~2 for each Yi to showthat T has a N(nμ, nσ2) distribution, andthusconcludethat Y has a N(μ, σ2~n) distribution.

(c) If Yi ∼ Pois(μi) for i = 1, ...,n, usethe mgf m(t) = exp[μi(et − 1)] for Yi to findthe distribution of T.

(d) Ifeach Yi ∼ binom(1, π), showthat mi(t) = [(1 − π) + πet]. Explainwhyabinom(n, π)

distribution has mgf m(t) = [(1 − π) + πet]n.

(e) If Yi ∼ binom(ni, πi), for i = 1, ...,n, underwhatcondition,ifany,does T haveabinomial distribution?

(f)The mgf of agammadistribution(2.10)near t = 0 is m(t) = [λ~(λ − t)]k. Ifeach Yi has an exponentialdistributionwithparameter λ, showthatthedistributionof T is gamma with shapeparameter k = n. Whathappenstothisshapeas n increases?