Energy operator

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

• 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

${\begin{aligned}f(x;\alpha )&={\frac {\sin \pi \alpha }{\pi }}x^{-\alpha }(1-x)^{\alpha -1}\\[6pt]\end{aligned}}$

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 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

• Arcsine

References

• Template:Eom

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