Arcsine distribution

Arcsine – bounded support
Parameters
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis

Arcsine
	Probability density function
	Cumulative distribution function
Parameters	none
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

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 $X\sim {\rm {Beta}}{\bigl (}{\tfrac {1}{2}},{\tfrac {1}{2}}{\bigr )}$ .

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

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

f(x;\alpha )={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}

is also a special case of the beta distribution with parameters ${\rm {Beta}}(1-\alpha ,\alpha )$ .

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 $\sin(U)$ , $\sin(2U)$ , $-\cos(2U)$ , $\sin(U+V)$ and $\sin(U-V)$ all have an ${\rm {Arcsine}}(-1,1)$ distribution.
If $X$ is the generalized arcsine distribution with shape parameter $\alpha$ supported on the finite interval [a,b] then ${\frac {X-a}{b-a}}\sim {\rm {Beta}}(1-\alpha ,\alpha )\$

References

Rogozin, B.A. (2001) [1994], "A/a013160", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

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	$-{\tfrac {3}{2}}$
Entropy	$\ln {\tfrac {\pi }{4}}$
MGF	$1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {2r+1}{2r+2}}\right){\frac {t^{k}}{k!}}$
CF	${}_{1}F_{1}({\tfrac {1}{2}};1;i\,t)\$

Arcsine – bounded support
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	$-{\tfrac {3}{2}}$