From: hrubin@stat.purdue.edu (Herman Rubin) Subject: Re: Gaussian Moments Date: 12 Dec 1999 20:57:53 -0500 Newsgroups: sci.math.research In article <3852D31E.E323BE80@ip.pt>, Manuel Costa wrote: >Hi all >Where can i found the gaussian moment expression centered about the >origin and expressed as average and variance function? I have this >result developed by myself: >m=(V+2)/2 if V is even >m=(V+1)/2 if V is odd > E(x^V)=V!*(Sum j=0 to m-1)1/2^j*1/(V-j)!*(V-j >j)*(V(x))^j*(E(x))^(V-2*j) >Note: (Sum j=0 to m-1) represents the sum from 0 to m-1 > (V-j j) is a combination >I need this result for a middle step, but since I cannot believe its >uniqueness, i need to know where can i found a book or a paper showing >it. Gaussian expression is so studied! This is easily shown in either of two ways. One way is write the random variable x as E(x)+y, and expand the expected value of x^n by the binomial theorem, noting that odd powers of y have expected value 0. The other is to use the fact that E(exp(tx)) = exp (t*E(x)+.5*t^2*V(x))), and to compute the n-th moment as the n-th derivative at 0. This is n! times the coefficient of t^n, which is obtained by multiplying the two series. This is essentially the same computation. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558