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.

with(Statistics):

MomentGeneratingFunction('Beta'(p, q), t);

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

MomentGeneratingFunction('Beta'(3, 5), 1/2);

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

MomentGeneratingFunction('Beta'(3, 5), 1/2, numeric);

$1.210195092$

T := Distribution(PDF = (t -> piecewise(t < 0, 0, t < 1, 6*t*(1-t), 0))):

X := RandomVariable(T):

MGF(X, u);

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

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