The easiest method to derive the moment-generating function of a general normal distribution

is to find the moment for a standard normal

and then use the formula for the linear transformation of a moment. Given

, we have a probability space.

Lemma: Let

be an absolutely continuous random variable whose moment-generating function is

. Then if

, then

Proof: Let a second random variable

. Then the moment generating function for

is

Therefore, if

, then

- this completes the lemma.

Consider

, which is standard normally distributed with mean 0 and variance 1, so that

The moment generating function for

is calculated by

1 2
Finally, consider

, which is normally distributed with mean

and variance

, so that

Recalling that

from our lemma, we have

Sorry, I know that you wanted the variables in terms of

instead of

but

is the convention that I am used to, haha...