11 - Taylor Expansion of the Moment Generating Function

Taylor Expansion of the Moment Generating Function

Suppose $M_X(t)$ is finite for $|t|\le t_0$ for some $t_0>0$ then

$$M_X(t)=\sum _{k=0}^{\infty }\frac{t^K\mathbb{E}[X^k]}{k!}\text{ for }|t|\le t_0$$

$$M_X^{(k)}(0)=\mathbb{E}[X^k]$$

Proof - Informal

$$\begin{array}{rl}{M}_X(t)& =\mathbb{E}[e^{tX}]\\ & =\mathbb{E}\left[1+tX+\frac{(tX{)}^2}{2!}+\frac{(tX{)}^3}{3!}+\cdots \right]\\ & =1+t\mathbb{E}[X]+\frac{t^2\mathbb{E}[X^2]}{2!}+\frac{t^3\mathbb{E}[X^3]}{3!}+\cdots \end{array}$$