LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) f

Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.