Moment Generating Function (mgf)

Definition

Let $X$ : Random variable

If

$$\exists h>0\ni E(e^{tX})<\infty ,\forall t\in (-h,h)$$

Then we say $M_X(t)=E(e^{tX})$ is the moment generating function (mgf) of $X$

Remark

Another way of defining mgf in terms of neighborhood is given below

Let $X$ : Random variable

If $E(e^{tX})$ exists and is finite in some neighborhood of $t=0$

Then we say $M_X(t)=E(e^{tX})$ is the moment generating function (mgf) of $X$

Most important property of mgf is that it uniquely identify distributions, as shown in the following theorem:

Related theorems

• Uniqueness of MGF