CentralMomentGeneratingFunction
CentralMomentGeneratingFunction[dist,t]
gives the central moment-generating function for the distribution dist as a function of the variable t.
CentralMomentGeneratingFunction[dist,{t1,t2,…}]
gives the central moment-generating function for the multivariate distribution dist as a function of the variables t1, t2, ….
Details
- CentralMomentGeneratingFunction[dist,t] is given by Expectation[Exp[t(x-μ)],xdist] where μ=Mean[dist].
- CentralMomentGeneratingFunction[dist, {t1,t2,…}] is equivalent to Expectation[Exp[t.(x-μ)],xdist] for vectors t and x and where μ=Mean[dist].
- The i
central moment can be extracted from a central moment-generating function cmgf through SeriesCoefficient[cmgf,{t,0,i}]i!.
Examples
open all close allBasic 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}]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, {xExponentialDistribution[Subscript[λ, 1]], yExponentialDistribution[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]Applications (3)
Find the cmgf of the sum of random variates:
CentralMomentGeneratingFunction[TransformedDistribution[x + y, {xErlangDistribution[2, Subscript[λ, 1]], yErlangDistribution[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 
it coincides with the central moment-generating function of ErlangDistribution:
% /. {Subscript[λ, 2] -> Subscript[λ, 1]}Confirm with TransformedDistribution:
TransformedDistribution[x + y, {xErlangDistribution[2, a], yErlangDistribution[3, a]}]Find the first few central moments of the sum of 
i.i.d. non-central 
random variates:
cmgf = CentralMomentGeneratingFunction[NoncentralChiSquareDistribution[ν, δ], t]^nTable[Limit[D[cmg, {t, k}], t -> 0], {k, 2, 4}]Illustrate the central limit theorem using ExponentialDistribution:
dist = ExponentialDistribution[λ];Find the cmgf of the exponential variate rescaled to have variance 
:
CentralMomentGeneratingFunction[TransformedDistribution[x / Sqrt[n Variance[dist]], xdist], t]Find the large 
limit of the cmgf of the sum of 
such variates:
Limit[% ^ n, n -> Infinity]Compare with the cmgf of the standard normal variate:
CentralMomentGeneratingFunction[NormalDistribution[], t]Properties & Relations (3)
The cmgf is the moment-generating function times 
:
𝒟 = ExponentialDistribution[λ];CentralMomentGeneratingFunction[𝒟, t]MomentGeneratingFunction[𝒟, t]Exp[-t Mean[𝒟]]Use SeriesCoefficient to find central moment 
:
SeriesCoefficient[CentralMomentGeneratingFunction[BetaDistribution[2, 3], t]r!, {t, 0, r}]//SimplifyCompare with CentralMoment:
CentralMoment[BetaDistribution[2, 3], r]Table[% == %%, {r, 0, 6}]CentralMomentGeneratingFunction is an exponential generating function for the sequence of central moments:
ExponentialGeneratingFunction[CentralMoment[ExponentialDistribution[a], r], r, t]CentralMomentGeneratingFunction[ExponentialDistribution[a], t]% - %%//FullSimplifyPossible Issues (2)
For some distributions with long tails, central moments of only several low orders are defined:
CentralMoment[StudentTDistribution[n], r]Correspondingly, CentralMomentGeneratingFunction is undefined:
CentralMomentGeneratingFunction[StudentTDistribution[n], t]CentralMomentGeneratingFunction is not always known in closed form:
CentralMomentGeneratingFunction[LogNormalDistribution[μ, σ], t]Use CentralMoment to evaluate particular central moments:
CentralMoment[LogNormalDistribution[μ, σ], 3]Neat Examples (1)
Visualize cmgf over complex plane for some univariate distributions:
dists = {NegativeBinomialDistribution[10, 2 / 3], PoissonDistribution[3], BorelTannerDistribution[5 / 6, 10], ExponentialDistribution[1], BirnbaumSaundersDistribution[1, 3], HyperbolicDistribution[2, 1, 1, 2]};Table[Plot3D[Re[CentralMomentGeneratingFunction[𝒟, x + I y]]//Evaluate, {x, -2, 2}, {y, -4, 4}, Mesh -> None, ImageSize -> 200, PlotLabel -> 𝒟], {𝒟, dists}]Text
Wolfram Research (2010), CentralMomentGeneratingFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/CentralMomentGeneratingFunction.html.
CMS
Wolfram Language. 2010. "CentralMomentGeneratingFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CentralMomentGeneratingFunction.html.
APA
Wolfram Language. (2010). CentralMomentGeneratingFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CentralMomentGeneratingFunction.html