MA2102

Probability and Statistics

Lecture-15

Moment Generating Function(MGF)

Let XX be a random variable defined on (Ω,F,P)(\Omega,\mathscr{F},P), and if E(etX)E(e^tX) exists in some neighbourhood of 00(i.e h<t<h-h<t<h, for some h>0h>0) then we say MGFMGF exists and we write as MX(t)=E(etX)M_X(t)=E(e^{tX}), h<t<h-h<t<h, h>0h>0

Example: Let XX be a discrete random variable with support SX={1,2,3,....}S_X=\{1,2,3,....\} and PMFPMF pX(n)=(13)(23)n1p_X(n)=\left(\frac{1}{3}\right)\left(\frac{2}{3}\right)^{n-1}, n=1,2,3,...n=1,2,3,...

n=1etnpX(n)=n=1etn(13)(23)n1\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 en=ennR|e^n|=e^{n}\forall n\in\mathbb{R} as en>0e^n>0)