Arcsine distribution

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Arcsine
Probability density function
Cumulative distribution function
Parameters	none
Support	$x \in [0,1]$
PDF	$f(x) = \frac{1}{\pi\sqrt{x(1-x)}}$
CDF	$F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right)$
Mean	$\frac{1}{2}$
Median	$\frac{1}{2}$
Mode	$x \in {0,1}$
Variance	$\tfrac{1}{8}$
Skewness	$0$
Ex. kurtosis	$-\tfrac{3}{2}$
MGF	$1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!}$
CF	${}_1F_1(\tfrac{1}{2}; 1; i\,t)\$

Contents

1 Standard distribution
2 Generalization
- 2.1 Arbitrary bounded support
- 2.2 Shape factor
3 Properties
4 Related distributions
5 See also
6 References

Standard distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function is

$F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x\right)=\frac{\arcsin(2x-1)}{\pi}+\frac{1}{2}$

for 0 ≤ x ≤ 1, and whose probability density function is

$f(x) = \frac{1}{\pi\sqrt{x(1-x)}}$

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if $X$ is the standard arcsine distribution then $X \sim {\rm Beta}(\tfrac{1}{2},\tfrac{1}{2}) \$

The arcsine distribution appears

in the Lévy arcsine law;
in the Erdős arcsine law;
as the Jeffreys prior for the probability of success of a Bernoulli trial.

Generalization

Arcsine – bounded support
Probability density function Need image
Cumulative distribution function Need image
Parameters	$-\infty < a < b < \infty \,$
Support	$x \in [a,b]$
PDF	$f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}}$
CDF	$F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)$
Mean	$\frac{a+b}{2}$
Median	$\frac{a+b}{2}$
Mode	$x \in {a,b}$
Variance	$\tfrac{1}{8}(b-a)^2$
Skewness	$0$
Ex. kurtosis	$-\tfrac{3}{2}$

Arbitrary bounded support

The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation

$F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)$

for a ≤ x ≤ b, and whose probability density function is

$f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}}$

on (a, b).

Shape factor

The generalized standard arcsine distribution on (0,1) with probability density function

$\begin{align} f(x;\alpha) & = \frac{\sin \pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1} \\[6pt] \end{align}$

is also a special case of the beta distribution with parameters $Beta(1 - α,α)$ .

Note that when $\alpha = \tfrac{1}{2}$ the general arcsine distribution reduces to the standard distribution listed above.

Properties

Arcsine distribution is closed under translation and scaling by a positive factor
- If $X \sim {\rm Arcsine}(a,b) \ \text{then } kX+c \sim {\rm Arcsine}(ak+c,bk+c)$
The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
- If $X \sim {\rm Arcsine}(-1,1) \ \text{then } X^2 \sim {\rm Arcsine}(0,1)$

Related distributions

If U and V are i.i.d uniform (−π,π) random variables, then $sin(U)$ , $sin(2 U)$ , $- cos(2 U)$ , $sin(U + V)$ and $sin(U - V)$ all have a standard arcsine distribution
If $X$ is the generalized arcsine distribution with shape parameter $α$ supported on the finite interval [a,b] then $\frac{X-a}{b-a} \sim {\rm Beta}(1-\alpha,\alpha) \$

References

Rogozin, B.A. (2001), "Arcsine distribution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=A/a013160

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Noncentral beta Irwin–Hall Kumaraswamy logit-normal raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

generalized inverse Gaussian

half-logistic

half-normal

Hotelling's T-squared

hyper-exponential

hypoexponential

inverse chi-squared (scaled-inverse-chi-squared)

noncentral chi-squared

Pareto

phase-type

Rayleigh

relativistic Breit–Wigner

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial multivariate Pólya negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse-Wishart matrix normal Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

v d e Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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Arcsine distribution