Let X1, 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...

Let X1, 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(e^sX) = λ/λ−s for s <>
(b)  Using (a) find the expected value E(Xi) and the variance Var(Xi).
(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .
(d) Consider a random variable X having Gamma(α, λ) distribution,
fX(x) = (λ^αx^α-1/Γ(α)) e−λx , x > 0
0 , otherwise
Show that the moment generating function of the random variable X is mX(s) =λ^α
1/(λ−s)^α for s < λ,="" where="" γ(α)="">
Γ(α) = (integral from 0 to inifity ) x^α−1e^−xdx.
(e)  What is the probability distribution of Y given in (c)? Explain your
answer.