The MomentGeneratingFunction function computes the moment generating function of the specified random variable at the specified point.

The first parameter can be a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]).

By default, all computations involving random variables are performed symbolically (see option numeric below).

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

numeric=truefalse -- By default, the moment generating function is computed using exact arithmetic. To compute the moment generating function numerically, specify the numeric or numeric = true option.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$\mathrm{MomentGeneratingFunction}\left(\'\mathrm{\Β}\'\left(p\,q\right)\,t\right)$

$\mathrm{hypergeom}\left(\left[p\right]\,\left[p\+q\right]\,t\right)$

$\mathrm{MomentGeneratingFunction}\left(\'\mathrm{\Β}\'\left(3\,5\right)\,\frac{1}{2}\right)$

$\mathrm{hypergeom}\left(\left[3\right]\,\left[8\right]\,\frac{1}{2}\right)$

$\mathrm{MomentGeneratingFunction}\left(\'\mathrm{\Β}\'\left(3\,5\right)\,\frac{1}{2}\,\mathrm{numeric}\right)$

$1.210195092$

$T≔\mathrm{Distribution}\left(\mathrm{PDF}\&equals;\left(t\&rarr;\mathrm{piecewise}\left(t<0\&comma;0\&comma;t<1\&comma;6t\left(1-t\right)\&comma;0\right)\right)\right)\&colon;$

$X≔\mathrm{RandomVariable}\left(T\right)\&colon;$

$\mathrm{MGF}\left(X\&comma;u\right)$

$\frac{6\left({\&ExponentialE;}^{u}u-2{\&ExponentialE;}^u\&plus;u\&plus;2\right)}{u^3}$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.