Suppose x is a gaussian 1 1 random variable and k is an

Question

Suppose X is a Gaussian (1, 1) random variable and K is an independent discrete random variable with PMF

Let X₁, X₂,... denote a sequence of iid random variables each with the same distribution as X.

(a) What is the MGF of K?

(b) What is the MGF of R = X₁ +···+ X_K? Note that R = 0 if K = 0.

(c) Find E[R] and Var[R].