Moment generating function

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In probability theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification of the random variable's 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 n-th moment about 0 is the 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...

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