A random variable X has a N(0,1) distribution. Use the moment generating function of X to compute (a) the third and (b) fourth moments of X, i.e., E(X³) and E(X4).

Let Z be a discrete random variable withE(Z)=0. Does it necessarily follow that E(Z3)=0? If yes, give a proof; if no, give a counterexample.

Let X have the pdf f (x) = ¿e-lxl, -0 < x < +o, find out the moment generating function of X. Show your work. Suppose X and Y are independent Poisson random variables, where X has mean |A and Y has mean v. Find out the moment generating function of X+Y and its mean value. Show your work.

Let X be a discrete random variable with P(X = -1) = .3, P(X = 1) = .45, P(X = 2) = .15 and P(X = 6) = .1. Find E(X).