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. 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.
Definition Let X {\displaystyle X} be a random variable with CDF F X {\displaystyle F_{X}} . The moment generating function (mgf) of X {\displaystyle X} (or F X {\displaystyle F_{X}} ), denoted by M X ( t ) {\displaystyle M_{X}(t)} , is
M X ( t ) = E [ e t X ] {\displaystyle M_{X}(t)=\operatorname {E} \left[e^{tX}\right]}
provided this expectation exists for t {\displaystyle t} in some open neighborhood of 0. That is, there is an h > 0 {\displaystyle h>0} such that for all t {\displaystyle t} satisfying − h < t < h {\displaystyle -h<t<h} , E [ e t X ] {\displaystyle \operatorname {E} \left[e^{tX}\right]} exists. If the expectation does not exist in an open 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 e t X {\displaystyle e^{tX}} . More generally, when X = ( X 1 , … , X n ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{n})^{\mathrm {T} }} , an n {\displaystyle n} -dimensional random vector, and t {\displaystyle \mathbf {t} } is a fixed vector, one uses t ⋅ X = t T X {\displaystyle \mathbf {t} \cdot \mathbf {X} =\mathbf {t} ^{\mathrm {T} }\mathbf {X} } instead of t X {\displaystyle tX} :
M X ( t ) := E [ e t T X ] . {\displaystyle M_{\mathbf {X} }(\mathbf {t} ):=\operatorname {E} \left[e^{\mathbf {t} ^{\mathrm {T} }\mathbf {X} }\right].}
M X ( 0 ) {\displaystyle M_{X}(0)} 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. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead. The moment generating function is so named because it can be used to find the moments of the distribution. The series expansion of e t X {\displaystyle e^{tX}} is
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