10 - Properties of the Moment Generating Function

Properties of the Moment Generating Function

Let $X$ be a random variable

1. Let $Y=aX+b$ for some $a,b\in \mathbb{R}$

$$M_Y(t)=e^{bt}{M}_X(at)$$

2. Let $X_1,\cdots ,X_n$ be independent random variables with mgfs $M_{X_1},\cdots ,M_{X_N}$ then
   Mgf of sum is defined by

$$M_{X_1+\cdots +X_n}(t)=M_{X_1}(t)\cdots M_{X_n}(t)$$