BUILT-IN MATHEMATICA SYMBOL

CentralMomentGeneratingFunction

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

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

,

, ....

MORE INFORMATION

CentralMomentGeneratingFunction is given by Expectation[Exp[t(x-)], xdist] where =Mean[dist].

CentralMomentGeneratingFunction is equivalent to Expectation[Exp[t.(x-)], xdist] for vectors t and x and where =Mean[dist].

The i central moment can be extracted from a central moment generating function cmgf through SeriesCoefficient[cmgf, {t, 0, i}]i!.

EXAMPLES

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Basic Examples (3)

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:

Confirm with TransformedDistribution:

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:

Properties & Relations (3)

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:

Possible Issues (2)

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: