Arcsine distribution

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Arcsine
Probability density function
Probability density function for the arcsine distribution
Cumulative distribution function
Cumulative distribution function for the arcsine distribution
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)\

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

Generalization[edit]

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}

Arbitrary bounded support[edit]

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 (ab).

Shape factor[edit]

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[edit]

  • 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)

Differential equation


\left\{2 (x-1) x f'(x)+(2 x-1) f(x)=0\right\}

Related distributions[edit]

  • 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 a standard arcsine 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) \

See also[edit]

References[edit]