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

Finally, consider , which is normally distributed with mean and variance , so that

Recalling that from our lemma, we have