View source for Arcsine distribution

{{MSC|60E99}}

[[Category:Distribution theory]]

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" />.

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

<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>

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.

====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>