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probability theory > probability distribution > arcsine distribution

mathematical statistics > probability distribution > arcsine distribution

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arcsine distribution

In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
${\\displaystyle F(x)={\ rac {2}{\\pi }}\\arcsin \\left({\\sqrt {x}}\ ight)={\ rac {\\arcsin(2x-1)}{\\pi }}+{\ rac {1}{2}}}$
for 0 ≤ x ≤ 1, and whose probability density function is
${\\displaystyle f(x)={\ rac {1}{\\pi {\\sqrt {x(1-x)}}}}}$
on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if ${\\displaystyle X}$ is an arcsine-distributed random variable, then ${\\displaystyle X\\sim {\ m {Beta}}{\igl (}{\ frac {1}{2}},{\ frac {1}{2}}{\igr )}}$ . By extension, the arcsine distribution is a special case of the Pearson type I distribution.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Arcsine_distribution)

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http://data.loterre.fr/ark:/67375/PSR-BL62N1W1-H

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