This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

MomentGeneratingFunction

 MomentGeneratingFunction gives the moment generating function for the symbolic distribution dist as a function of the variable t. MomentGeneratingFunctiongives the moment generating function for the multivariate symbolic distribution dist as a function of the variables , , ....
• The i moment can be extracted from a moment generating function mgf through SeriesCoefficient[mgf, {t, 0, i}]i!.
Compute the moment generating function (mgf) for a continuous univariate distribution:
The mgf for a univariate discrete distribution:
The mgf for a multivariate distribution:
Compute the moment generating function (mgf) for a continuous univariate distribution:
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The mgf for a univariate discrete distribution:
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The mgf for a multivariate distribution:
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 Scope   (4)
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:
 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:
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 standard normal distribution:
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:
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:
New in 8