Question

What is the moment generating function of a Gaussian distribution?

Answer 1

If `X` is Normal (Gaussian) with mean `mu` and standard deviation `sigma`, its moment generating function is: `m_{X}(t)=e^{mu t+(sigma^[2} t^{2})/2`.

Explanation:

Deriving this formula requires a calculation of the integral:

`m_{X}(t)=E[e^{tX}]=1/(sigma sqrt(2pi})\int_{-infty}^{infty} e^{tx}e^{-1/2(x-mu)^2/sigma^{2}}\ dt`.

This can be done by expanding `-1/2(x-mu)^2/sigma^{2}`, completing the square, and using properties of exponents, but the details are not very pleasant. It can also be done by a substitution `z=(x-mu)/sigma`. The details are found at the pdf file found at the following website: derivation of mgf of Normal