mgf

In a) I've problem understanding if I should compute the mgf of the function f(x) or mgf of expected value?

I am given the MGF of IID RVs as e^{\lambda(e^t-1)} and am supposed to find the MGF of their sample mean of n such RVs. I have am arriving at M_n(t) = e^{n\lambda(e^t - 1)+1} This doesn't appear to be a valid MGF and there is another part to the question where it asks for the \lim_{n...

E(X^r)=(r+1)!* (2^r). we have to find MGF of random variable X.

Let X1 and X2 be two independent random variables. Let X1 and Y = X1 + X2 be χ2(r1) (Chi Square) and χ2(r), respectively, where r1 < r. (a) Find the mgf of X2. (b) What is its distribution?

I think it's something to do with the Non central Chisquared distribution but I'm not sure how to use it and derive the mgf of W? Would someone show be step by step guidance?

The discrete random variable X has probability function p(x)=4/(5^X+1) X=0, 1, 2,... Derive the MGF of X and use it to find E(X) and V(X). I have managed to get this far: Mx(t) = Σ(e^tX)(4/(5^X+1)) e^(tX) = 1 + tX + (t^2/2!)X^2 + (t^3/3!)X^3 + ... So Mx(t) = Σ(4/(5^X+1)) +...

Let X1, . . . , Xn be IID Exponential(?) random variables. Use mgfs to find the distributionofY =X1 +...Xn.

Hello, I am having trouble understanding the maths between E(e^{tx}) where I get a summation from 0 to \infty and the m.g.f for Poisson distribution. The notes I have are to break out e^{-\lambda}/e^{-\lambda e^t} (Now, don't get me wrong - I DO get how to make this last expression easier on...