---
tags: Probability
---

# Further Topics on Random Variables: Transforms (Moment Generating Functions)

## Transforms

* Also called **moment generating functions** of rvs.
* The **transform** of the distribution of rv $X$ is a function $M_X(s)$ of a free parameter $s$
$$
M_X(s) = E[e^{sX}]
$$
* If $X$ is discrete
$$
M_X(s) = \sum_xe^{sx}p_X(x)
$$
* If $X$ is continuous
$$
M_X(s) = \int_{-\infty}^{\infty}e^{sx}f_X(x) \ \mathrm{d}x
$$