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, … .
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, … .
Details
- MomentGeneratingFunction is also called a raw moment-generating function.
- MomentGeneratingFunction [dist,t] is equivalent to Expectation [Exp [t x],xdist].
- MomentGeneratingFunction [dist,{t1,t2,…}] is equivalent to Expectation [Exp [t.x],xdist] for vectors t and x.
- The i^(th) moment can be extracted from a moment-generating function mgf through SeriesCoefficient [mgf,{t,0,i}]i!.
Examples
open all close allBasic 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)
History
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_2025_momentgeneratingfunction, author="Wolfram Research", title="{MomentGeneratingFunction}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/MomentGeneratingFunction.html}", note=[Accessed: 17-November-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: 17-November-2025]}