Английская Википедия:Arcsine distribution

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Шаблон:Short description Шаблон:Probability distribution</math> |

 cdf        =<math>F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x \right)</math> |
 mean       =<math>\frac{1}{2}</math> |
 median     =<math>\frac{1}{2}</math> |
 mode       =<math>x \in \{0,1\}</math> |
 variance   =<math>\tfrac{1}{8}</math> |
 skewness   =<math>0</math>|
 kurtosis   =<math>-\tfrac{3}{2}</math>|
 entropy    =<math>\ln \tfrac{\pi}{4}</math> |
 mgf        =<math>1  +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{2r+1}{2r+2} \right) \frac{t^k}{k!}</math>|
 char       =<math>{}_1F_1(\tfrac{1}{2}; 1; i\,t)\ </math>|

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

<math>F(x) = \frac{2}{\pi}\arcsin\left(\sqrt x\right)=\frac{\arcsin(2x-1)}{\pi}+\frac{1}{2}</math>

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

<math>f(x) = \frac{1}{\pi\sqrt{x(1-x)}}</math>

on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if <math>X</math> is an arcsine-distributed random variable, then <math>X \sim {\rm Beta}\bigl(\tfrac{1}{2},\tfrac{1}{2}\bigr)</math>. 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.

Generalization

Шаблон:Probability distribution</math> |

 cdf        =<math>F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)</math> |
 mean       =<math>\frac{a+b}{2}</math> |
 median     =<math>\frac{a+b}{2}</math> |
 mode       =<math>x \in {a,b}</math> |
 variance   =<math>\tfrac{1}{8}(b-a)^2</math> |
 skewness   =<math>0</math>|
 kurtosis   =<math>-\tfrac{3}{2}</math>|
 entropy    = |
 mgf        = |
 char       = |

}}

Arbitrary bounded support

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

<math>F(x) = \frac{2}{\pi}\arcsin\left(\sqrt \frac{x-a}{b-a} \right)</math>

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

<math>f(x) = \frac{1}{\pi\sqrt{(x-a)(b-x)}}</math>

on (a, b).

Shape factor

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

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

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

Note that when <math>\alpha = \tfrac{1}{2}</math> 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 <math>X \sim {\rm Arcsine}(a,b) \ \text{then } kX+c \sim {\rm Arcsine}(ak+c,bk+c) </math>
• The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
  • If <math>X \sim {\rm Arcsine}(-1,1) \ \text{then } X^2 \sim {\rm Arcsine}(0,1) </math>
• The coordinates of points uniformly selected on a circle of radius <math>r</math> centered at the origin (0, 0), have an <math>{\rm Arcsine}(-r,r)</math> distribution
  • For example, if we select a point uniformly on the circumference, <math>U \sim {\rm Uniform}(0,2\pi r)</math>, we have that the point's x coordinate distribution is <math>r \cdot \cos(U) \sim {\rm Arcsine}(-r,r) </math>, and its y coordinate distribution is <math display="inline">r \cdot \sin(U) \sim {\rm Arcsine}(-r,r) </math>

Characteristic function

The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as <math>{}_1F_1(\tfrac{1}{2}; 1; i\,t)\ </math>.

Related distributions

• If U and V are i.i.d uniform (−π,π) random variables, then <math>\sin(U)</math>, <math>\sin(2U)</math>, <math>-\cos(2U)</math>, <math>\sin(U+V)</math> and <math>\sin(U-V)</math> all have an <math>{\rm Arcsine}(-1,1)</math> distribution.
• If <math>X</math> is the generalized arcsine distribution with shape parameter <math>\alpha</math> supported on the finite interval [a,b] then <math>\frac{X-a}{b-a} \sim {\rm Beta}(1-\alpha,\alpha) \ </math>
• If X ~ Cauchy(0, 1) then <math>\tfrac{1}{1+X^2}</math> has a standard arcsine distribution

Further reading

• Шаблон:Eom