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Concept
Moment-generating function
Summary
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 functions or cumulative distribution functions. 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 moments: the nth moment about 0 is the nth 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 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.
Let be a random variable with CDF . The moment generating function (mgf) of (or ), denoted by , is
provided this expectation exists for in some neighborhood of 0. That is, there is an such that for all in , exists. If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist.
In other words, the moment-generating function of X is the expectation of the random variable . More generally, when , an -dimensional random vector, and is a fixed vector, one uses instead of :
always exists and is equal to 1. However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely.
Official source
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