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Standard distribution 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 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{align} f(x;\alpha) & = \frac{\sin \pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1} \\[6pt] \end{align}
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 ( , ) 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 References fr:Loi arc sinus
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