BUILT-IN MATHEMATICA SYMBOL

CumulantGeneratingFunction

gives the cumulant generating function for the symbolic distribution dist as a function of the variable t.

gives the cumulant generating function for the multivariate symbolic distribution dist as a function of the variables

,

, ....

MORE INFORMATION

CumulantGeneratingFunction is given by Log[MomentGeneratingFunction[dist, t]].

CumulantGeneratingFunction is given by Log[MomentGeneratingFunction[dist, {t₁, t₂, ...}]].

The i cumulant can be extracted from a cumulant generating function cgf through SeriesCoefficient[cgf, {t, 1, i}]i!.

EXAMPLES

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Basic Examples (3)

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:

In[1]:=

Out[1]=

The cgf for a univariate discrete distribution:

In[1]:=

Out[1]=

The cgf for a multivariate distribution:

In[1]:=

Out[1]=

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:

Properties & Relations (3)

Exponential of CumulantGeneratingFunction gives MomentGeneratingFunction:

CumulantGeneratingFunction is an exponential generating function for the sequence of cumulants:

Use CumulantGeneratingFunction directly:

Cumulant

is equivalent to

:

Use SeriesCoefficient formulation:

Possible Issues (2)

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: