Hi
Can someone please explain the process in creating the moment generating function for the normal distribution? i can do it for the other special distributions but not the normal.
thanks.
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