Properties of a Moment-Generating Function

Topics: Moment-Generating Function - Probability

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(theorem)

1. $M_Y(t)=e^{bt}{M}_X(at)$, where $Y=aX+b$

2. $M_X(t)={\sum }_{n=0}^{\infty }\frac{t^n}{n!}\mathbb{E}[X^n]$

3. If $X$ and $Y$ are independent random variables, then $M_{X+Y}(t)=M_X(t)M_Y(t)$.

   • But $M_{X+Y}(t)=M_X(t)M_Y(t)\text{centernot}⟹X\perp \perp Y$

4. If $M_X(t)=M_Y(t)\forall t$, then $X$ and $Y$ have the same distribution (the same density function)

5. Moment-Generating Function of the Sum of the Random Variables in a Random Sample