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Moyal distribution

Quick view

The Moyal distribution is an asymmetric real-line law that appears as an approximation for some energy-loss shapes and as a normal transform. It has a longer right tail than a symmetric bell.

If you are coming from another distribution

It is related to the Landau energy-loss function and to normal transforms. It is not a universal Poisson approximation or a count distribution.

History and terminology

J. E. Moyal obtained the shape in 1955 while studying ionization fluctuations and energy loss. Its analytic expression became familiar in particle physics and radiation.

A familiar situation

In detectors, a Moyal-like asymmetric density can approximate deposited-energy response with a large-loss tail. For general data, an equally strong physical justification is needed.

Fitting with care

Do not use it for counts merely because its history mentions Poisson processes; the Moyal variable is continuous.

An analytic approximation to energy loss

In 1955 J. E. Moyal derived a universal form for ionization fluctuations produced by a fast particle. The density carrying his name supplies a manageable analytic approximation to the Landau shape used for energy loss in thin layers.

Moyal is not exactly Landau. Both have a peak and a long right tail, but their tails and quantiles differ, which matters in detector calibration and extrapolation. In the usual parameterization, mu is the mode rather than the mean; the latter is shifted by known constants. The relevant source is Moyal’s 1955 paper on ionization fluctuations, not his later work on approximately Poisson distributions.

Decision guide

A good candidate when: the process is compatible with continuous energy loss having a peak and a long high-value tail.

Compare it with: Landau for detector physics and Lognormal for generic skewness. Moyal is a specific analytic approximation, not a universal right-tail model.

References

Moyal Distribution: equations and calculator

Distribution defintion

X∼Moyal(μ,σ)X\sim\mathrm{Moyal}\left(\mu,\sigma\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

μ∈R,σ∈R+\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−P(12,e−z(x)2)=1−erf(exp⁡(−0.5z(x))2)F_{X}\left(x\right)=1-\text{P}\left(\frac{1}{2},\frac{e^{-z(x)}}{2}\right)=1-\mathrm{erf}\left(\frac{\exp\left(-0.5z(x)\right)}{\sqrt{2}}\right)

Probability density function

fX(x)=12πexp⁡(−12(z(x)+e−z(x)))f_{X}\left(x\right)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(z(x)+e^{-z(x)}\right)\right)

Percent point function/Sample

FX−1(u)=μ+σln⁡[Φ−1((1−u2)2)]=μ+σln⁡[2P−1(12,1−u)]F^{-1}_{X}\left(u\right)=\mu+\sigma\ln\left[\Phi^{-1}\left(\left(\frac{1-u}{2}\right)^{2}\right)\right]=\mu+\sigma\ln\left[2\text{P}^{-1}\left(\frac{1}{2},1-u\right)\right]

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=μ+σ(ln⁡(2)+γ)\mathrm{Mean}(X)=\mu'_{1}=\mu+\sigma(\ln(2)+\gamma)

Parametric variance

Variance(X)=μ2′−μ1′2=σ2(π22)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\sigma^{2}\left(\frac{\pi^{2}}{2}\right)

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=282ζ(3)π3\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{28\sqrt{2}\zeta(3)}{\pi^{3}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=7\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=7

Parametric median

Median(X)=μ+σln⁡[2P−1(12,12)]\mathrm{Median}(X)=\mu+\sigma\ln\left[2\text{P}^{-1}\left(\frac{1}{2},\frac{1}{2}\right)\right]

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
erf(x):Error function\mathrm{erf}(x):\text{Error function}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649
ζ(3):Apeˊry’s constant=1.2020569031\zeta(3):\text{Apéry's constant}=1.2020569031