Arcsine distribution

{{Probability distribution |

name =Arcsine|

type =density|

pdf_image =Image:Arcsin density.svg|

cdf_image =Image:Arcsin cdf.svg|

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 = $0$ |

kurtosis = $-\tfrac{3}{2}$ |

entropy = $\ln \tfrac{\pi}{4}$ |

mgf = $1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!}$ |

char = $e^{i\frac{t}{2}}J_0(\frac{t}{2})$ |

}}

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:

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

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 an arcsine-distributed random variable, then $X \sim {\rm Beta}\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr)$ . By extension, the arcsine distribution is a special case of the Pearson type I distribution.

The arcsine distribution appears in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial.{{cite conference |last1=Overturf |first1=Drew |last2=Buchanan |first2=Kristopher |last3=Jensen |first3=Jeffrey |last4=Wheeland |first4=Sara |last5=Huff |first5=Gregory |display-authors=1 |year=2017 |title=Investigation of beamforming patterns from volumetrically distributed phased arrays |conference=MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM) |pages=817–822 |doi=10.1109/MILCOM.2017.8170756 |isbn=978-1-5386-0595-0 }}{{cite journal |first1=K. |last1=Buchanan |first2=J. |last2=Jensen |first3=C. |last3=Flores-Molina |first4=S. |last4=Wheeland |first5=G. H. |last5=Huff |display-authors=1 |title=Null Beamsteering Using Distributed Arrays and Shared Aperture Distributions |journal=IEEE Transactions on Antennas and Propagation |volume=68 |issue=7 |pages=5353–5364 |year=2020 |doi=10.1109/TAP.2020.2978887 |bibcode=2020ITAP...68.5353B }} The arcsine probability density is a distribution that appears in several random-walk fundamental theorems. In a fair coin toss random walk, the probability for the time of the last visit to the origin is distributed as an (U-shaped) arcsine distribution.{{cite book|last=Feller|first=William|title=An Introduction to Probability Theory and Its Applications, Vol. 2|year=1971|publisher=Wiley|isbn=978-0471257097|url=https://archive.org/details/introductiontopr00fell}}{{cite book |last=Feller |first=William |title=An Introduction to Probability Theory and Its Applications |volume=1 |edition=3rd |year=1968 |publisher=Wiley |isbn=978-0471257080}} 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 not N. 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).

Generalization

{{Probability distribution |

name =Arcsine – bounded support|

type =density|

pdf_image = |

cdf_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 = $0$ |

kurtosis = $-\tfrac{3}{2}$ |

entropy = |

mgf = |

char = $e^{it\frac{b+a}{2}}J_0(\frac{b-a}{2}t)$ |

}}

=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 arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
If $X \sim {\rm Arcsine}(-1,1) \ \text{then } X^2 \sim {\rm Arcsine}(0,1)$
The coordinates of points uniformly selected on a circle of radius $r$ centered at the origin (0, 0), have an ${\rm Arcsine}(-r,r)$ distribution
For example, if we select a point uniformly on the circumference, $U \sim {\rm Uniform}(0,2\pi r)$ , we have that the point's x coordinate distribution is $r \cdot \cos(U) \sim {\rm Arcsine}(-r,r)$ , and its y coordinate distribution is $r \cdot \sin(U) \sim {\rm Arcsine}(-r,r)$

Characteristic function

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

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) \$
If X ~ Cauchy(0, 1) then $\tfrac{1}{1+X^2}$ has a standard arcsine distribution