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

# CumulantGeneratingFunction

 CumulantGeneratingFunction gives the cumulant generating function for the symbolic distribution dist as a function of the variable t. CumulantGeneratingFunction gives the cumulant generating function for the multivariate symbolic distribution dist as a function of the variables , , ....
• The i cumulant can be extracted from a cumulant generating function cgf through SeriesCoefficient[cgf, {t, 1, i}]i!.
Compute a cumulant generating function (cgf) for a continuous univariate distribution:
The cgf for a univariate discrete distribution:
The cgf for a multivariate distribution:
Compute a cumulant generating function (cgf) for a continuous univariate distribution:
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The cgf for a univariate discrete distribution:
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The cgf for a multivariate distribution:
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 Scope   (4)
Compute the cgf for a formula distribution:
Find the cgf for a function of random variates:
Compute the cgf for data distribution:
Find the cgf for a truncated distribution:
 Applications   (4)
The cumulant generating function of a difference of two independent random variables is equal to the difference of their cumulant generating functions:
Illustrate the central limit theorem:
Find the cumulant 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:
Find the Esscher premium for insuring against losses following GammaDistribution:
Compare with the definition:
Construct a Bernstein-Chernoff bound for the survival function :
Large approximation of the bound:
CumulantGeneratingFunction is an exponential generating function for the sequence of cumulants:
Cumulant is equivalent to :
Use SeriesCoefficient formulation:
For some distributions with long tails, cumulants of only several low orders are defined:
Correspondingly, CumulantGeneratingFunction is undefined:
CumulantGeneratingFunction is not always known in closed form:
Use Cumulant to find cumulants directly:
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