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October 11, 2020

find the moment generating function of Z and use it to find the first six moments of Z.

(Auxiliary Functions and Moments). For a random variable X, the characteristic function E(e^itX) always exists since e^itX is a bounded random variable (with complex values). The moment generating function E(e^tx) only exists if X has moments of all orders and those moments do not grow too fast.

a) If Z is Gaussian with mean 0 and variance 1, find the moment generating function of Z and use it to find the first six moments of Z. In particular, show that E(Z⁴) = 3, a fact that we will need later. [Hint: Series expansion can be easier than differentiation.]

b) If W = Z⁴ , then W has moments of all orders, since Z has all moments. Show nevertheless that W does not have a moment generating function.

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