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Statistics

 MomentGeneratingFunction
 compute the moment generating function

 Calling Sequence MomentGeneratingFunction(X, t, options) MGF(X, t, options)

Parameters

 X - algebraic; random variable or distribution t - algebraic; point options - (optional) equation of the form numeric=value; specifies options for computing the moment generating function of a random variable

Description

 • 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]).

Computation

 • 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.

Options

 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.

Examples

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

Compute the moment generating function of the beta distribution with parameters p and q.

 > $\mathrm{MomentGeneratingFunction}\left('\mathrm{Β}'\left(p,q\right),t\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{p}\right]{,}\left[{p}{+}{q}\right]{,}{t}\right)$ (1)

Use numeric parameters.

 > $\mathrm{MomentGeneratingFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2}\right)$
 ${\mathrm{hypergeom}}{}\left(\left[{3}\right]{,}\left[{8}\right]{,}\frac{{1}}{{2}}\right)$ (2)
 > $\mathrm{MomentGeneratingFunction}\left('\mathrm{Β}'\left(3,5\right),\frac{1}{2},\mathrm{numeric}\right)$
 ${1.210195092}$ (3)

Define new distribution.

 > $T≔\mathrm{Distribution}\left(\mathrm{PDF}=\left(t→\mathrm{piecewise}\left(t<0,0,t<1,6t\left(1-t\right),0\right)\right)\right):$
 > $X≔\mathrm{RandomVariable}\left(T\right):$
 > $\mathrm{MGF}\left(X,u\right)$
 $\frac{{6}{}\left({{ⅇ}}^{{u}}{}{u}{-}{2}{}{{ⅇ}}^{{u}}{+}{u}{+}{2}\right)}{{{u}}^{{3}}}$ (4)

References

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