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MomentGeneratingFunction—Wolfram Language Documentation

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BUILT-IN SYMBOL

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, … .

Details Examplesopen allclose all 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) 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.htmlBibTeX @misc{reference.wolfram_2025_momentgeneratingfunction, author="Wolfram Research", title="{MomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}", note=[Accessed: 12-July-2025 ]}BibLaTeX @online{reference.wolfram_2025_momentgeneratingfunction, organization={Wolfram Research}, title={MomentGeneratingFunction}, year={2010}, url={https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}, note=[Accessed: 12-July-2025 ]}

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