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 $X\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(\frac{1}{2},\frac{1}{2})$

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

$\begin{array}{rl}f(x;\alpha )& =\frac{\sin \pi \alpha }{\pi }{x}^{-\alpha }(1-x{)}^{\alpha -1}\\ \end{array}$

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

Note that when $\alpha =\frac{1}{2}$ 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 $X\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(a,b)\text{then }kX+c\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(ak+c,bk+c)$
• The square of an arc sine distribution over (-1, 1) has arc sine distribution over (0, 1)
  • If $X\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(-1,1)\text{then }{X}^2\sim \mathrm{A}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{e}(0,1)$

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 $\frac{X-a}{b-a}\sim \mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}(1-\alpha ,\alpha )$

See also

• Arcsine

References

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