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Arcsine distribution

The idea in one sentence

The Arcsine distribution is a Beta law with both parameters equal to one half, transformed to [a,b]. Its density is largest near both endpoints and smallest in the middle: it is intentionally U-shaped rather than bell-shaped.

Support and interpretation

The support is bounded. A squared-sine transformation of a Uniform variable generates this law and makes its endpoint concentration intuitive.

Its place in the family

It is a special case of the Beta family. It also arises as a limiting law in occupation-time and last-maximum problems, although those stochastic-process results do not mean that every bounded dataset should have an arcsine shape.

An applied reading

It is a useful reference when observations on [a,b] accumulate near both boundaries. If the endpoints are caused by censoring, rounding, or instrument limits, the U-shape may be an artefact rather than a property of the phenomenon.

Do not fit it to values exactly at the limits without deciding whether the support is closed, open, or produced by rounding.

Spending a long time near the extremes

The U shape is not a decorative oddity. One of Lévy’s arcsine laws says that the fraction of time Brownian motion spends above zero follows this distribution. Contrary to first intuition, a path has substantial probability of spending nearly the whole interval on one side.

On the unit interval, Arcsine is Beta with both shapes equal to one half. This puts it opposite the Semicircular law, which concentrates in the centre and corresponds to shapes of three halves. Infinite density peaks at the endpoints are not probability atoms: under the continuous model, the probability of observing an endpoint exactly remains zero.

Decision guide

A good candidate when: the variable is bounded and observations genuinely accumulate near both endpoints, leaving the centre relatively sparse.

Compare it with: a free Beta model. Arcsine fixes both shape parameters at 1/2; if accumulation differs between endpoints, Beta is usually a more honest description.

References

  • SciPy reference: scipy.stats.arcsine — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. Wiley.
  • Lévy, P. (1939). Sur certains processus stochastiques homogènes. Compositio Mathematica, 7, 283–339.

Arcsine Distribution: equations and calculator

Distribution defintion

X∼Arcsine(a,b)X\sim\mathrm{Arcsine}\left(a,b\right)

Distribution domain

x∈(a,b)x\in\left(a,b\right)

Parameters domain and parameters constraints

a∈R,b∈R,a<ba\in\mathbb{R},b\in\mathbb{R},a < b

Cumulative distribution function

FX(x)=2πarcsin⁡(x−ab−a)F_{X}(x)=\frac{2}{\pi}\arcsin\left(\sqrt\frac{x-a}{b-a}\right)

Probability density function

fX(x)=1π(x−a)(b−x)f_{X}(x)=\frac{1}{\pi\sqrt{(x-a)(b-x)}}

Percent point function/Sample

FX−1(u)=a+(b−a)×sin⁡2(π2u)F_{X}^{-1}\left(u\right)=a+\left(b-a\right)\times \sin^{2}\left(\frac{\pi}{2}u\right)

Non-central parametric moments

μ~k′=E[X~k]=∫01xkfX~(x)dx=1πBeta(12,k+12)=(2k−1)!!2kk!\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{1}{\pi}\text{Beta}\left(\frac{1}{2},k+\frac{1}{2}\right)=\frac{\left(2k-1\right)!!}{2^{k}k!}

Parametric mean

Mean(X)=a+μ~1′(b−a)=a+12(b−a)\mathrm{Mean}(X)=a+\tilde{\mu}'_{1}\left(b-a\right)= a+\frac{1}{2}\left(b-a\right)

Parametric variance

Variance(X)=(b−a)2×(μ~2′−μ~1′2)=(b−a)28\mathrm{Variance}(X)=\left(b-a\right)^{2}\times (\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\frac{\left(b-a\right)^{2}}{8}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=0\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3−32\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3-\frac{3}{2}

Parametric median

Median(X)=a+(b−a)×sin⁡2(π4)\mathrm{Median}(X)=a+\left(b-a\right)\times \sin^{2}\left(\frac{\pi}{4}\right)

Parametric mode

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

Additional information and definitions

X~∼Arcsine(0,1)\tilde{X}\sim \mathrm{Arcsine}\left(0,1\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}