Moment-Generating Function

Topics: Probability - Moment

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(definition)

$$M_X(t):=\mathbb{E}\left[e^{tX}\right]$$

…with $t\in (-h,h)$ given $h>0$. This implies that $M_X(t)$ is differentiable at $t=0$.

Do note that not all distributions of probability have a moment-generating function.

Moment-generating functions have several properties.

Discrete Case §

(theorem, from the law of the unconscious statistician)

When $X$ is a discrete random variable. then:

$$M_X(t)=\sum _{x\in R_X}{e}^{tx}{f}_X(x)$$

Continuous Case §

(theorem, from the law of the unconscious statistician)

When $X$ is an absolutely continuous random variable, then:

$$M_X(t)={\int }_{\mathbb{R}}{e}^{tx}{f}_X(x)dx$$

$n$th moment §

(theorem)

In general, we have that:

$$E[X^n]={\frac{d^n{M}_X(t)}{d{t}^n}|}_{t=0}$$

That is, the $n$th (non-central) moment of $X$ is given by the $n$th derivative of $M_X(t)$ evaluated at $t=0$.