# MomentGeneratingFunction

MomentGeneratingFunction[dist,t]

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

MomentGeneratingFunction[dist,{t1,t2,}]

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

The mgf for a univariate discrete distribution:

The mgf for a multivariate distribution:

## Scope(5)

Compute the moment-generating function (mgf) for a formula distribution:

Find the mgf for a function of a random variate:

Find the mgf for a data distribution:

Compute the mgf for a censored distribution:

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

## Applications(3)

Find the moment-generating function of the sum of random variates:

Check that it is equal to the product of generating functions:

When it coincides with the mgf of BinomialDistribution:

Confirm with TransformedDistribution:

Reconstruct the PDF of a positive real random variate from its moment-generating function:

Check the result:

Illustrate the central limit theorem on the example of PoissonDistribution:

Find the moment-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:

## Properties & Relations(5)

MomentGeneratingFunction is equivalent to Expectation of :

MomentGeneratingFunction is an exponential generating function for the sequence of moments:

Use SeriesCoefficient to find moment :

Use Moment directly:

MomentGeneratingFunction is a LaplaceTransform for positive random variables:

MomentGeneratingFunction is a ZTransform for discrete positive random variates:

## Possible Issues(2)

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

Correspondingly, MomentGeneratingFunction is undefined:

Analytic continuation of CharacteristicFunction can sometimes be defined:

MomentGeneratingFunction is not always known in closed form:

Use Moment to evaluate particular moments:

## Neat Examples(1)

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_momentgeneratingfunction, author="Wolfram Research", title="{MomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_momentgeneratingfunction, organization={Wolfram Research}, title={MomentGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}, note=[Accessed: 19-July-2024 ]}