({{, (RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction])[dist,t] 
gives the central moment-generating function for the distribution dist as a function of the variable t. }, {, (RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction])[dist,{(Subscript[t, 1]),(Subscript[t, 2]),…}]
gives the central moment-generating function for the multivariate distribution dist as a function of the variables (Subscript[t, 1]), (Subscript[t, 2]), …. }})

OpenersNotesSection(Spacer1)Details

(RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction])[dist,t] is given by (RefLink[Expectation, paclet:ref/Expectation])[(RefLink[Exp, paclet:ref/Exp])[t(x-μ)],x\[Distributed]dist] where μ=(RefLink[Mean, paclet:ref/Mean])[dist].

(RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction])[dist, {(Subscript[t, 1]),(Subscript[t, 2]),…}] is equivalent to (RefLink[Expectation, paclet:ref/Expectation])[(RefLink[Exp, paclet:ref/Exp])[t.(x-μ)],x\[Distributed]dist] for vectors t and x and where μ=(RefLink[Mean, paclet:ref/Mean])[dist].

The i(^th) central moment can be extracted from a central moment-generating function cmgf through (RefLink[SeriesCoefficient, paclet:ref/SeriesCoefficient])[cmgf,{t,0,i}]i!.

OpenersPrimaryExamplesSection(Spacer1)Examples  (16)

OpenersExampleSection0(Spacer1)Basic Examples  (3)

Compute a central moment-generating function (cmgf) for a univariate continuous distribution:

CentralMomentGeneratingFunction[NormalDistribution[μ,σ],t]

The cmgf for a univariate discrete distribution:

CentralMomentGeneratingFunction[PoissonDistribution[μ],t]

The cmgf for a multivariate distribution:

CentralMomentGeneratingFunction[BinormalDistribution[ρ],{t1,t2}]

OpenersExampleSection1(Spacer1)Scope  (5)

The central moment-generating function for a formula distribution:

CentralMomentGeneratingFunction[ProbabilityDistribution[((2 (1+x) (E^(-x)))/(3 (Sqrt[x]) (Sqrt[π]))),{x,0,∞}],t]

Find the cmgf for a function of random variates:

CentralMomentGeneratingFunction[TransformedDistribution[x y,{x\[Distributed]ExponentialDistribution[(Subscript[λ, 1])],y\[Distributed]ExponentialDistribution[(Subscript[λ, 2])]}],t]

Find the cmgf for data distribution:

hdist=HistogramDistribution[ExampleData[{"Statistics","FatigueLifeFailures"}]]

cmgf=CentralMomentGeneratingFunction[hdist,t]

Find the cmgf for censored distribution:

CentralMomentGeneratingFunction[CensoredDistribution[{-3,3},NormalDistribution[]],t]

Find the cmgf for the slice distribution of a random process:

CentralMomentGeneratingFunction[PoissonProcess[μ][s],t]

OpenersExampleSection2(Spacer1)Applications  (3)

Find the cmgf of the sum of random variates:

CentralMomentGeneratingFunction[TransformedDistribution[x+y,{x\[Distributed]ErlangDistribution[2,(Subscript[λ, 1])],y\[Distributed]ErlangDistribution[3,(Subscript[λ, 2])]}],t]

Alternatively, compute the product of cmgfs of summands:

CentralMomentGeneratingFunction[ErlangDistribution[2,(Subscript[λ, 1])],t]CentralMomentGeneratingFunction[ErlangDistribution[3,(Subscript[λ, 2])],t]

When (Subscript[λ, 1])==(Subscript[λ, 2]) it coincides with the central moment-generating function of (RefLink[ErlangDistribution, paclet:ref/ErlangDistribution]):

%/.{(Subscript[λ, 2])->(Subscript[λ, 1])}

Confirm with (RefLink[TransformedDistribution, paclet:ref/TransformedDistribution]):

TransformedDistribution[x+y,{x\[Distributed]ErlangDistribution[2,a],y\[Distributed]ErlangDistribution[3,a]}]

Find the first few central moments of the sum of n i.i.d. non-central (χ^2) random variates:

cmgf=((CentralMomentGeneratingFunction[NoncentralChiSquareDistribution[ν,δ],t])^n)

Table[Limit[D[cmg,{t,k}],t->0],{k,2,4}]

Illustrate the central limit theorem using (RefLink[ExponentialDistribution, paclet:ref/ExponentialDistribution]):

dist=ExponentialDistribution[λ];

Find the cmgf of the exponential variate rescaled to have variance n:

CentralMomentGeneratingFunction[TransformedDistribution[x/(Sqrt[n Variance[dist]]),x\[Distributed]dist],t]

Find the large n limit of the cmgf of the sum of n such variates:

Limit[%^n,n->Infinity]

Compare with the cmgf of the standard normal variate:

CentralMomentGeneratingFunction[NormalDistribution[],t]

OpenersExampleSection3(Spacer1)Properties & Relations  (3)

The cmgf is the moment-generating function times exp(-t μ):

𝒟=ExponentialDistribution[λ];

CentralMomentGeneratingFunction[𝒟,t]

MomentGeneratingFunction[𝒟,t]Exp[-t Mean[𝒟]]

Use (RefLink[SeriesCoefficient, paclet:ref/SeriesCoefficient]) to find central moment (Subscript[(Overscript[μ, ∼]), r]):

SeriesCoefficient[CentralMomentGeneratingFunction[BetaDistribution[2,3],t]r!,{t,0,r}]//Simplify

Compare with (RefLink[CentralMoment, paclet:ref/CentralMoment]):

CentralMoment[BetaDistribution[2,3],r]

Table[%==%%,{r,0,6}]

(RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction]) is an exponential generating function for the sequence of central moments:

ExponentialGeneratingFunction[CentralMoment[ExponentialDistribution[a],r],r,t]

CentralMomentGeneratingFunction[ExponentialDistribution[a],t]

%-%%//FullSimplify

OpenersExampleSection4(Spacer1)Possible Issues  (2)

For some distributions with long tails, central moments of only several low orders are defined:

CentralMoment[StudentTDistribution[n],r]

Correspondingly, (RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction]) is undefined:

CentralMomentGeneratingFunction[StudentTDistribution[n],t]

(RefLink[CentralMomentGeneratingFunction, paclet:ref/CentralMomentGeneratingFunction]) is not always known in closed form:

CentralMomentGeneratingFunction[LogNormalDistribution[μ,σ],t]

Use (RefLink[CentralMoment, paclet:ref/CentralMoment]) to evaluate particular central moments:

CentralMoment[LogNormalDistribution[μ,σ],3]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[CentralMoment, paclet:ref/CentralMoment])  ▪  (RefLink[Expectation, paclet:ref/Expectation])  ▪  (RefLink[CharacteristicFunction, paclet:ref/CharacteristicFunction])  ▪  (RefLink[MomentGeneratingFunction, paclet:ref/MomentGeneratingFunction])  ▪  (RefLink[CumulantGeneratingFunction, paclet:ref/CumulantGeneratingFunction])  ▪  (RefLink[FactorialMomentGeneratingFunction, paclet:ref/FactorialMomentGeneratingFunction])  ▪  (RefLink[GeneratingFunction, paclet:ref/GeneratingFunction])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Statistical Distribution Functions, paclet:guide/StatisticalDistributionFunctions])}, {▪(RefLinkPlain[Statistical Moments and Generating Functions, paclet:guide/StatisticalMomentsAndGeneratingFunctions])}})}})