MA2102

Probability and Statistics

Lecture-15

Moment Generating Function(MGF)

Let $X$ be a random variable defined on $(\Omega,\mathscr{F},P)$, and if $E(e^tX)$ exists in some neighbourhood of $0$(i.e $-h<t<h$, for some $h>0$) then we say $MGF$ exists and we write as $M_X(t)=E(e^{tX})$, $-h<t<h$, $h>0$

Example: Let $X$ be a discrete random variable with support $S_X=\{1,2,3,....\}$ and $PMF$ $p_X(n)=\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^{n-1}$, $n=1,2,3,...$

$\sum_{n=1}^{\infty}|e^{tn}|p_X(n)=\sum_{n=1}^{\infty}e^{tn}\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^{n-1}$ ($\because$ $|e^n|=e^{n}\forall n\in\mathbb{R}$ as $e^n>0$)