Arcsine
Probability density function
Cumulative distribution function
Parameters	none
Support	${\displaystyle x\in (0,1)}$
PDF	${\displaystyle f(x)=\frac{1}{\pi \sqrt{x(1-x)}}}$
CDF	${\displaystyle F(x)=\frac{2}{\pi }\arcsin \left(\sqrt{x}\right)}$
Quantile	${\displaystyle F^{-1}(x)=\sin {\left(\frac{\pi x}{2}\right)}^2}$
Mean	${\displaystyle \frac{1}{2}}$
Median	${\displaystyle \frac{1}{2}}$
Mode	${\displaystyle x\in \{0,1\}}$
Variance	${\displaystyle \frac{1}{8}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle -\frac{3}{2}}$
Entropy	${\displaystyle \log \frac{\pi }{4}}$
MGF	${\displaystyle 1+\sum _{k=1}^{\mathrm{\infty }}\left(\prod _{r=0}^{k-1}\frac{2r+1}{2r+2}\right)\frac{t^k}{k!}}$
CF	${\displaystyle e^{i\frac{t}{2}}{J}_0(\frac{t}{2})}$

Inprobability theory, thearcsine distribution is theprobability distribution whosecumulative distribution function involves thearcsine and thesquare root:

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

for 0 ≤ x ≤ 1, and whoseprobability density function is

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

on (0, 1). The standard arcsine distribution is a special case of thebeta distribution withα = β = 1/2. That is, if${\displaystyle X}$ is an arcsine-distributed random variable, then${\displaystyle X\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\frac{1}{2},\frac{1}{2})}$. By extension, the arcsine distribution is a special case of thePearson type I distribution.

The arcsine distribution appears in theLévy arcsine law, in theErdős arcsine law, and as theJeffreys prior for the probability of success of aBernoulli trial.^[1][2] The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin tossrandom walk, the probability for the time of the last visit to the origin is distributed as an (U-shaped)arcsine distribution.^[3][4] In a two-player fair-coin-toss game, a player is said to be in the lead if the random walk (that started at the origin) is above the origin. The most probable number of times that a given player will be in the lead, in a game of length 2N, is notN. On the contrary,N is the least likely number of times that the player will be in the lead. The most likely number of times in the lead is 0 or 2N (following the arcsine distribution).

Arcsine – bounded support
Parameters
Support	${\displaystyle x\in (a,b)}$
PDF	${\displaystyle f(x)=\frac{1}{\pi \sqrt{(x-a)(b-x)}}}$
CDF	${\displaystyle F(x)=\frac{2}{\pi }\arcsin \left(\sqrt{\frac{x-a}{b-a}}\right)}$
Quantile	${\displaystyle F^{-1}(x)=\left(b-a\right)\sin {\left(\frac{\pi x}{2}\right)}^2+a}$
Mean	${\displaystyle \frac{a+b}{2}}$
Median	${\displaystyle \frac{a+b}{2}}$
Mode	${\displaystyle x\in a,b}$
Variance	${\displaystyle \frac{1}{8}(b-a{)}^2}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle -\frac{3}{2}}$
Entropy	${\displaystyle \log \left(\pi \frac{b-a}{4}\right)}$
CF	${\displaystyle e^{it\frac{b+a}{2}}{J}_0(\frac{b-a}{2}t)}$

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

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

fora ≤ x ≤ b, and whoseprobability density function is

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

on (a, b).

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

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

is also a special case of thebeta distribution with parameters${\displaystyle \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(1-\alpha ,\alpha )}$.

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

• Arcsine distribution is closed under translation and scaling by a positive factor
  • If${\displaystyle X\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(a,b)\text{then }kX+c\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(ak+c,bk+c)}$
• The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
  • If${\displaystyle X\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-1,1)\text{then }{X}^2\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(0,1)}$
• The coordinates of points uniformly selected on a circle of radius${\displaystyle r}$ centered at the origin (0, 0), have an${\displaystyle \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-r,r)}$ distribution
  • For example, if we select a point uniformly on the circumference,${\displaystyle U\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,2\pi r)}$, we have that the point's x coordinate distribution is${\displaystyle r\cdot \cos (U)\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-r,r)}$, and its y coordinate distribution is$r\cdot \sin (U)\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-r,r)$

The characteristic function of the generalized arcsine distribution is a zero orderBessel function of the first kind, multiplied by a complex exponential, given by${\displaystyle e^{it\frac{b+a}{2}}{J}_0(\frac{b-a}{2}t)}$. For the special case of${\displaystyle b=-a}$, the characteristic function takes the form of${\displaystyle J_0(bt)}$.

• If U and V arei.i.duniform (−π,π) random variables, then${\displaystyle \sin (U)}$,${\displaystyle \sin (2U)}$,${\displaystyle -\cos (2U)}$,${\displaystyle \sin (U+V)}$ and${\displaystyle \sin (U-V)}$ all have an${\displaystyle \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-1,1)}$ distribution.
• If${\displaystyle X}$ is the generalized arcsine distribution with shape parameter${\displaystyle \alpha }$ supported on the finite interval [a,b] then${\displaystyle \frac{X-a}{b-a}\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(1-\alpha ,\alpha )}$
• IfX ~ Cauchy(0, 1) then${\displaystyle \frac{1}{1+X^2}}$ has a standard arcsine distribution

• Rogozin, B.A. (2001) [1994],"Arcsine distribution",Encyclopedia of Mathematics,EMS Press

• Continuous distributions

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• Short description matches Wikidata