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CentralMomentGeneratingFunction

CentralMomentGeneratingFunction
gives the central moment generating function for the symbolic distribution dist as a function of the variable t.
CentralMomentGeneratingFunction
gives the central moment generating function for the multivariate symbolic distribution dist as a function of the variables , , ....
  • The i^(th) 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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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:
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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