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A [[Probability measure|probability measure]] on the real line whose density is zero outside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131601.png" /> and is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131602.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131603.png" />. The corresponding distribution function is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131604.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131605.png" />.
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{{TEX|done}}
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{{MSC|60E99}}
  
The generalized arcsine distribution is employed together with the arcsine distribution. To the generalized arcsine distribution corresponds the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131606.png" /> with density
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[[Category:Distribution theory]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131607.png" /></td> </tr></table>
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A [[Probability measure|probability measure]] on the real line whose density is zero outside the interval $(0,1)$ and is $(\sqrt{x(1-x)})^{-1}/\pi$ if $0<x<1$. The corresponding distribution function is equal to $(2/\pi)\arcsin\sqrt x$ for $0\leq x\leq1$.
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131608.png" />. The density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a0131609.png" /> coincides with the density of the arcsine distribution. The generalized arcsine distribution is a special case of the [[Beta-distribution|beta-distribution]]. The first-order moment of the generalized arcsine distribution is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a01316010.png" />, and its variance is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013160/a01316011.png" />. The arcsine distribution and the generalized arcsine distribution occur in the study of the fluctuations of random walks, in renewal theory (cf. [[Arcsine law|Arcsine law]]), and are used in mathematical statistics as special cases of the beta-distribution.
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The generalized arcsine distribution is employed together with the arcsine distribution. To the generalized arcsine distribution corresponds the distribution function $F_\alpha(x)$ with density
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$$f_\alpha(x)=\begin{cases}\frac{\sin\pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1}&\text{if }0<x<1,\\0&\text{if }x\leq0,x\geq1,\end{cases}$$
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if $0<a<1$. The density $f_{1/2}(x)$ coincides with the density of the arcsine distribution. The generalized arcsine distribution is a special case of the [[Beta-distribution|beta-distribution]]. The first-order moment of the generalized arcsine distribution is $1-\alpha$, and its variance is $(1-\alpha)\alpha/2$. The arcsine distribution and the generalized arcsine distribution occur in the study of the fluctuations of random walks, in renewal theory (cf. [[Arcsine law|Arcsine law]]), and are used in mathematical statistics as special cases of the beta-distribution.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1957–1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.G. Kendall,   A. Stuart,   "The advanced theory of statistics. Distribution theory" , '''3. Design and analysis''' , Griffin  (1969)</TD></TR></table>
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|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''' , Wiley (1957–1971)
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|valign="top"|{{Ref|KS}}|| M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" {{MR|0246399}} {{ZBL|}}
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Latest revision as of 12:50, 11 October 2014

2010 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

A probability measure on the real line whose density is zero outside the interval $(0,1)$ and is $(\sqrt{x(1-x)})^{-1}/\pi$ if $0<x<1$. The corresponding distribution function is equal to $(2/\pi)\arcsin\sqrt x$ for $0\leq x\leq1$.

The generalized arcsine distribution is employed together with the arcsine distribution. To the generalized arcsine distribution corresponds the distribution function $F_\alpha(x)$ with density

$$f_\alpha(x)=\begin{cases}\frac{\sin\pi\alpha}{\pi}x^{-\alpha}(1-x)^{\alpha-1}&\text{if }0<x<1,\\0&\text{if }x\leq0,x\geq1,\end{cases}$$

if $0<a<1$. The density $f_{1/2}(x)$ coincides with the density of the arcsine distribution. The generalized arcsine distribution is a special case of the beta-distribution. The first-order moment of the generalized arcsine distribution is $1-\alpha$, and its variance is $(1-\alpha)\alpha/2$. The arcsine distribution and the generalized arcsine distribution occur in the study of the fluctuations of random walks, in renewal theory (cf. Arcsine law), and are used in mathematical statistics as special cases of the beta-distribution.

References

[F] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1957–1971)
[KS] M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" MR0246399
How to Cite This Entry:
Arcsine distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arcsine_distribution&oldid=18892
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article