<P> 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 . </P> <P> 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 . </P> <P> 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 . </P> <P> 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 . </P>

How to find moment generating function of normal distribution