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Statistics

  

CumulantGeneratingFunction

  

compute the cumulant generating function

 

Calling Sequence

Parameters

Description

Computation

Options

Examples

References

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).

• 

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

Examples

withStatistics:

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

CumulantGeneratingFunction'Β'p,q,t

lnhypergeomp,p+q,t

(1)

Use numeric parameters.

CumulantGeneratingFunction'Β'3,5,12

lnhypergeom3,8,12

(2)

CumulantGeneratingFunction'Β'3,5,12,numeric

0.1907815797

(3)

Define new distribution.

TDistributionPDF&equals;t&rarr;piecewiset<0&comma;0&comma;t<1&comma;6t1t&comma;0&colon;

XRandomVariableT&colon;

CGFX&comma;u

ln6&ExponentialE;uu2&ExponentialE;u&plus;u&plus;2u3

(4)

References

  

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

See Also

Statistics

Statistics[Computation]

Statistics[Distributions]

Statistics[RandomVariables]

 


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