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DISTRIBUTIONS / CONTINUOUS / SEMICIRCULAR

Semicircular distribution

The idea in one sentence

The Semicircular distribution puts mass under a semicircle and is bounded and symmetric. loc sets the centre and R the radius, which also determines the possible interval.

Support and interpretation

Support is [loc-R, loc+R]. Density vanishes at the edges and peaks at the centre, clearly separating it from Arcsine.

Its place in the family

It is the Wigner distribution and belongs to random-matrix theory rather than ordinary sums of normals. Compared with Uniform it concentrates the centre; compared with Arcsine it avoids endpoint mass.

An applied reading

In mathematical physics it describes the global spectral density of certain large matrices. In applied data it can be a geometric model, but the spectral interpretation does not transfer automatically.

Do not use it as a universal model for bounded symmetric data: its shape comes from a specific spectral structure.

Eigenvalues fill a semicircle

Wigner discovered that eigenvalues of large random symmetric matrices, after suitable normalization, fill a semicircular density. The law became an entry point to random matrix theory and models for complex quantum spectra.

On a rescaled interval, the distribution is a symmetric Beta with shapes of three halves. Its density peaks in the centre and falls to zero at the edges, exactly opposite to Arcsine with shapes of one half. Even moments of the standard version are connected with Catalan numbers. It can fit generic bounded data, but the matrix story requires many eigenvalues and assumptions about their matrix, not merely a rounded curve.

Decision guide

A good candidate when: a geometric or random-matrix construction yields a symmetric density that vanishes smoothly at two finite endpoints.

Compare it with: a rescaled symmetric Beta and Uniform. Semicircular imposes a very specific curvature and is rarely the first choice for bounded data without a generating theory.

References

Semicircular Distribution: equations and calculator

Distribution defintion

X∼Semicircular(Loc,R)X\sim\mathrm{Semicircular}\left(\text{Loc},R\right)

Distribution domain

x∈[Loc,∞)x\in\left[\text{Loc},\infty\right)

Parameters domain and parameters constraints

Loc∈R,R∈R+\text{Loc}\in\mathbb{R},R\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=12+z(x)R2−z(x)2πR2+arcsin⁡ ⁣(z(x)R)πF_{X}\left(x\right)=\frac12+\frac{z(x)\sqrt{R^2-z(x)^2}}{\pi R^2}+\frac{\arcsin\!\left(\frac{z(x)}{R}\right)}{\pi}

Probability density function

fX(x)=2πR2 R2−z(x)2f_{X}\left(x\right)=\frac2{\pi R^2}\,\sqrt{R^2-z(x)^2}

Percent point function/Sample

FX−1(u)=Loc+R×(2I−1(u,1.5,1.5)−1)F^{-1}_{X}\left(u\right)=\text{Loc}+R\times (2I^{-1}\left(u,1.5,1.5\right)-1)

Non-central parametric moments

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

Parametric mean

Mean(X)=μ1′=Loc\mathrm{Mean}(X)=\mu'_{1}=\text{Loc}

Parametric variance

Variance(X)=μ2′−μ1′2=R24\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{R^2}{4}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=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}}=2

Parametric median

Median(X)=Loc\mathrm{Median}(X)=\text{Loc}

Parametric mode

Mode(X)=Loc\mathrm{Mode}(X)=\text{Loc}

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
R:Scale parameterR:\text{Scale parameter}
z(x)=x−Locz\left(x\right)= x-\text{Loc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}