Moment-generating function

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the moment-generating function of a real-valued <a href="/facts/Random_variable/TwTBXnLT">random variable</a> is an alternative specification of its <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a>. Thus, it provides the basis of an alternative route to analytical results compared with working directly with <a href="/facts/Probability_density_function/zvfybna4">probability density functions</a> or <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution functions</a>. 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-<a href="/facts/Generating_function/K7UVjnjL">generating function</a> can be used to compute a distribution’s <a href="/facts/Moment_(mathematics)/GL7vJ1nb">moments</a>: the n-th moment about 0 is the n-th derivative of the moment-generating function, evaluated at 0.
In addition to real-valued distributions (univariate distributions), moment-generating functions can 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 <a href="/facts/Characteristic_function_(probability_theory)/rl3uAGKM">characteristic function</a>. 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.

Moment-generating function open-in-new

Moment-generating function