2.5c Moment Generating Functions

Moment Generating Function (MGF)

A moment generating function (MGF) encodes all of the moments $\mathbb{E}[X^n]$ of a random variable into a single function, making it easier to study its distributional properties. Formally, the moment-generating function of a random variable $X$ is defined as
$M_X(t)=\mathbb{E}[e^{tX}]=\{\begin{array}{ll}\sum _{x\in \mathcal{X}}{e}^{tx}{p}_X(x)& x\text{ discrete},\\ {\int }_{-\infty }^{\infty }{e}^{tx}{f}_X(x)dx& x\text{ continuous}.\end{array}$
The MGF of $X$ can then be used to compute the $n$-th moment of $X$, such that
$\frac{d^n}{d{x}^n}{M}_X(t){|}_{x=0}=\mathbb{E}[X^n].$

Properties

(Linearity) If $$\begin{gathered} Y=\alpha \\ X+\beta \end{gathered}$$, then $M_Y(t)=e^{t\beta }{M}_X(\alpha x)$.

(Independence) If $X,Y$ are independent, then $M_{X+Y}(t)=M_X(t)M_Y(t)$.