Moment Generating Functions

Definition

The moment generating function (MGF), or Laplace Transform, of is defined by

where varies over the real numbers.

When the MGF is well defined, it can be shown that we can interchange the operations of differentiation and expectation. This leads to

By taking derivatives, we conclude that . This gives us a method for computing the moments of a distribution.

Lemma (Properties of the MFG)

1. If , then
2. If are independent and , Then , where is the MGF of .

Theorem

Let and be random variables. If for all is an open interval around 0, then .

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 3.6.