What are the multiple correlations of three sets of predictors and overall state of health? The first set of predictors contains demographic variables (age and years of education). The second….
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(eitX) always exists since eitX is a bounded random variable (with complex values). The moment generating function E(etx) 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.