# CumulantGeneratingFunction

CumulantGeneratingFunction[dist,t]

gives the cumulant-generating function for the distribution dist as a function of the variable t.

CumulantGeneratingFunction[dist,{t1,t2,}]

gives the cumulant-generating function for the multivariate distribution dist as a function of the variables t1, t2, .

# Examples

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## Basic Examples(3)

Compute a cumulant-generating function (cgf) for a continuous univariate distribution:

The cgf for a univariate discrete distribution:

The cgf for a multivariate distribution:

## Scope(5)

Compute the cgf for a formula distribution:

Find the cgf for a function of random variates:

Compute the cgf for data distribution:

Find the cgf for a truncated distribution:

Find the cgf for the slice distribution of a random process:

## Applications(5)

The cumulant-generating function of a difference of two independent random variables is equal to the sum of their cumulant-generating functions with oppositive sign arguments:

Illustrate the central limit theorem:

Find the cumulant-generating function for the standardized random variate:

Find the moment-generating function for the sum of standardized random variates rescaled by :

Find the large limit:

Compare with the moment-generating function of a standard normal distribution:

Find the Esscher premium for insuring against losses following GammaDistribution:

Compare with the definition:

Construct a BernsteinChernoff bound for the survival function :

Large approximation of the bound:

Construct Daniel's saddle point approximation to PDF of VarianceGammaDistribution:

Find the saddle point associated with the argument of probability density function :

Select the solution that is valid for all real , including the origin:

The approximation is constructed using the cumulant-generating function at the saddle point:

Find the normalization constant:

Compare the approximation to the exact density:

## Properties & Relations(3)

Exponential of CumulantGeneratingFunction gives MomentGeneratingFunction:

CumulantGeneratingFunction is an exponential generating function for the sequence of cumulants:

Use CumulantGeneratingFunction directly:

Cumulant is equivalent to :

Use SeriesCoefficient formulation:

## Possible Issues(2)

For some distributions with long tails, cumulants of only several low orders are defined:

Correspondingly, CumulantGeneratingFunction is undefined:

CumulantGeneratingFunction is not always known in closed form:

Use Cumulant to find cumulants directly:

Wolfram Research (2010), CumulantGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.

#### Text

Wolfram Research (2010), CumulantGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.

#### CMS

Wolfram Language. 2010. "CumulantGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html.

#### APA

Wolfram Language. (2010). CumulantGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html

#### BibTeX

@misc{reference.wolfram_2023_cumulantgeneratingfunction, author="Wolfram Research", title="{CumulantGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html}", note=[Accessed: 29-February-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_cumulantgeneratingfunction, organization={Wolfram Research}, title={CumulantGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/CumulantGeneratingFunction.html}, note=[Accessed: 29-February-2024 ]}