Suppose that the Random X ~ N(u*t,sigma^2*t). How do you use Mathematica to generate moment-generating e^X ?

Unhappily, it is very difficult to paste Mathematica code here on Yahoo! Answers.

Well, on Mathematica 7, you will need to use the definition of moment-generating function and be aware that Mathematica has an entity for each of several useful statistical distributions. The Normal Distribution is represented by NormalDistribution[mean,standard deviation]. There is also a method to obtain the probability distribution function (PDF). I have put it all together with some reasonable assumptions.

I will suggest you to try this (it is also on http://www.pastey.net/134871 where you can download it or even copy and paste it into Mathematica - but before copying you should toggle the line numbers off):

(* Mathematica code starts here *)

M[z_] =

Assuming[u\[Element]Reals

&&

t>0

&&

\[Sigma]>0 ,

\!\(

\*SubsuperscriptBox[\(\[Integral]\),

\(-\[Infinity]\),

\(\[Infinity]\)]

\(Exp[z\ x]

PDF[NormalDistribution[

u\ t,

\[Sigma]\

\*SqrtBox[\(t\)]], x]

\[DifferentialD]x\)\) ]

R[z_] := Log[M[z]]

R'[0]

R''[0]

(* Mathematica code ends here *)