# CentralMomentGeneratingFunction

CentralMomentGeneratingFunction[dist,t]

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

CentralMomentGeneratingFunction[dist,{t1,t2,}]

gives the central moment-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 central moment-generating function (cmgf) for a univariate continuous distribution:

The cmgf for a univariate discrete distribution:

The cmgf for a multivariate distribution:

## Scope(5)

The central moment-generating function for a formula distribution:

Find the cmgf for a function of random variates:

Find the cmgf for data distribution:

Find the cmgf for censored distribution:

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

## Applications(3)

Find the cmgf of the sum of random variates:

Alternatively, compute the product of cmgfs of summands:

When it coincides with the central moment-generating function of ErlangDistribution:

Confirm with TransformedDistribution:

Find the first few central moments of the sum of i.i.d. non-central random variates:

Illustrate the central limit theorem using ExponentialDistribution:

Find the cmgf of the exponential variate rescaled to have variance :

Find the large limit of the cmgf of the sum of such variates:

Compare with the cmgf of the standard normal variate:

## Properties & Relations(3)

The cmgf is the moment-generating function times :

Use SeriesCoefficient to find central moment :

Compare with CentralMoment:

CentralMomentGeneratingFunction is an exponential generating function for the sequence of central moments:

## Possible Issues(2)

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

Correspondingly, CentralMomentGeneratingFunction is undefined:

CentralMomentGeneratingFunction is not always known in closed form:

Use CentralMoment to evaluate particular central moments:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2022_centralmomentgeneratingfunction, author="Wolfram Research", title="{CentralMomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CentralMomentGeneratingFunction.html}", note=[Accessed: 08-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_centralmomentgeneratingfunction, organization={Wolfram Research}, title={CentralMomentGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/CentralMomentGeneratingFunction.html}, note=[Accessed: 08-June-2023 ]}