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# System prepended metadata

title: 'Further Topics on Random Variables: Transforms (Moment Generating Functions)'
tags: [Probability]

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tags: Probability
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# 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
$$
M_X(s) = \int_{-\infty}^{\infty}e^{sx}f_X(x) \ \mathrm{d}x
$$

