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Rayleigh distribution

The idea in one sentence

The Rayleigh distribution describes the magnitude of a two-dimensional vector whose components are independent centred normals. It is the natural amplitude law when in-phase and quadrature contributions are Gaussian noise.

Support and interpretation

Support begins at gamma. After subtracting that floor, the variable is nonnegative and sigma controls radial spread; the standard case uses gamma=0.

Its place in the family

It is Chi with two degrees of freedom and a shape-two Weibull case. Rice adds a deterministic component and describes amplitude when a dominant signal is present.

An applied reading

In communications, Rayleigh represents multipath without a dominant path. In oceanography it can be a reference for linear wave amplitudes, though large nonlinear waves can depart from it.

Do not use Rayleigh for amplitudes with a coherent signal component without comparing Rice or Nakagami.

Two Gaussian noises become one amplitude

Take two centred, independent Normal components with equal scale. The distance of their point from the origin follows Rayleigh. Lord Rayleigh encountered this shape while studying sums of vibrations with random phases.

Squared amplitude is Exponential after rescaling, a relationship that simplifies power models in radar and communications. Rayleigh is also Chi with two degrees of freedom and the Rice case without a deterministic component. If component scales differ or they are correlated, amplitude is no longer Rayleigh. In a wireless channel, accepting the model amounts to claiming diffuse scattering with no dominant line of sight.

Decision guide

A good candidate when: after subtracting gamma, the observation is the magnitude of two independent centred Gaussian components with equal variance.

Compare it with: Rice when a deterministic component exists and Nakagami for greater empirical flexibility. The square of a Rayleigh variable should exhibit Exponential structure.

References

  • SciPy reference: scipy.stats.rayleigh — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Rayleigh, Lord (1880). On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Philosophical Magazine, 10(60), 73–78.
  • Papoulis, A. (1965). Probability, Random Variables, and Stochastic Processes. McGraw-Hill.

Rayleigh Distribution: equations and calculator

Distribution defintion

X∼Rayleigh(γ,σ)X\sim\mathrm{Rayleigh}\left(\gamma,\sigma\right)

Distribution domain

x∈[γ,∞)x\in\left[\gamma,\infty\right)

Parameters domain and parameters constraints

γ∈R,σ∈R+\gamma\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−e−z(x)2/2F_{X}\left(x\right)=1-e^{-z(x)^{2}/2}

Probability density function

fX(x)=z(x)×e−z(x)2/2/σf_{X}\left(x\right)=z(x)\times e^{-z(x)^{2}/2}/\sigma

Percent point function/Sample

FX−1(u)=γ+σ−2log⁡(1−u)F^{-1}_{X}\left(u\right)=\gamma+\sigma\sqrt{-2\log\left(1-u\right)}

Non-central parametric moments

μk′=E[X~k]=∫0∞xkfX~(x)dx=2kΓ(k2+1)\mu'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\sqrt{2^{k}}\Gamma\left(\frac{k}{2}+1\right)

Parametric mean

Mean(X)=γ+σ⋅μ1′=γ+σπ2\mathrm{Mean}(X)=\gamma+\sigma\cdot\mu'_{1}=\gamma+\sigma\sqrt{\frac{\pi}{2}}

Parametric variance

Variance(X)=σ2(μ2′−μ1′2)=σ24−π2\mathrm{Variance}(X)=\sigma^{2}(\mu'_{2}-\mu'^{2}_{1})=\sigma^{2}\frac{4-\pi}{2}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(π−3)π(4−π)3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2\left(\pi-3\right)\sqrt{\pi}}{\left(4-\pi\right)^{3/2}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3+24π−6π2−16(4−π)2\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}}=3+\frac{24\pi-6\pi^{2}-16}{\left(4-\pi\right)^{2}}

Parametric median

Median(X)=γ+σ−2log⁡(12)\mathrm{Median}(X)=\gamma+\sigma\sqrt{-2\log\left(\frac{1}{2}\right)}

Parametric mode

Mode(X)=γ+σ\mathrm{Mode}(X)=\gamma+\sigma

Additional information and definitions

X~∼Rayleigh(0,1)\tilde{X}\sim\mathrm{Rayleigh}\left(0,1\right)
γ:Location parameter\gamma:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−γ)/σz\left(x\right)=\left(x-\gamma\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}