For a random variable , the moment-generating function is

  • If is discrete,
  • If is continuous,

For example, if ,

A moment-generating function can uniquely determine a distribution.

Use the Moment-generating Function to Calculate the sum of random variables

If the we have iid distributions , the moment-generating function of the sum is the moment-generating function of individual distribution raise to n-th power:

And we can get back a new distribution since moment generating functions uniquely determine distribution.