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Statistics

 CumulantGeneratingFunction
 compute the cumulant generating function

 Calling Sequence CumulantGeneratingFunction(X, t, options) CGF(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 cumulant generating function of a random variable

Description

 • The CumulantGeneratingFunction function computes the cumulant 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).

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 cumulant generating function is computed using exact arithmetic. To compute the cumulant generating function numerically, specify the numeric or numeric = true option.

Examples

 > with(Statistics):

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

 > CumulantGeneratingFunction('Beta'(p, q), t);
 ${\mathrm{ln}}{}\left({\mathrm{hypergeom}}{}\left(\left[{p}\right]{,}\left[{p}{+}{q}\right]{,}{t}\right)\right)$ (1)

Use numeric parameters.

 > CumulantGeneratingFunction('Beta'(3, 5), 1/2);
 ${\mathrm{ln}}{}\left({\mathrm{hypergeom}}{}\left(\left[{3}\right]{,}\left[{8}\right]{,}\frac{{1}}{{2}}\right)\right)$ (2)
 > CumulantGeneratingFunction('Beta'(3, 5), 1/2, numeric);
 ${0.1907815797}$ (3)

Define new distribution.

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

References

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