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
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...