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

CentralMomentGeneratingFunction

 CentralMomentGeneratingFunction gives the central moment generating function for the symbolic distribution dist as a function of the variable t. CentralMomentGeneratingFunctiongives the central moment generating function for the multivariate symbolic distribution dist as a function of the variables , , ....
• The i central moment can be extracted from a central moment generating function cmgf through SeriesCoefficient[cmgf, {t, 0, i}]i!.
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:
Compute a central moment generating function (cmgf) for a univariate continuous distribution:
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The cmgf for a univariate discrete distribution:
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The cmgf for a multivariate distribution:
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 Scope   (4)
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:
 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:
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:
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:
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:
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