Editing Moment-generating function (section)

{{Short description|Concept in probability theory and statistics}}
In [[probability theory]] and [[statistics]], the '''moment-generating function''' of a real-valued [[random variable]] is an alternative specification of its [[probability distribution]]. Thus, it provides the basis of an alternative route to analytical results compared with working directly with [[probability density function]]s or [[cumulative distribution function]]s. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.

As its name implies, the moment-[[generating function]] can be used to compute a distribution’s [[Moment (mathematics)|moments]]: the {{mvar|n}}-th moment about 0 is the {{mvar|n}}-th derivative of the moment-generating function, evaluated at 0.

In addition to univariate real-valued distributions, moment-generating functions can also be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.

The moment-generating function of a real-valued distribution does not always exist, unlike the [[Characteristic function (probability theory)|characteristic function]]. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.