# Difference between revisions of "Arcsine distribution"

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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" />. | 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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====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> | <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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## Revision as of 15:42, 7 January 2012

[ 2010 Mathematics Subject Classification MSN: 60E99 | MSCwiki: 60E99 ]

A probability measure on the real line whose density is zero outside the interval and is if . The corresponding distribution function is equal to for .

The generalized arcsine distribution is employed together with the arcsine distribution. To the generalized arcsine distribution corresponds the distribution function with density

if . The density 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 , and its variance is . 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

[1] | W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1957–1971) |

[2] | M.G. Kendall, A. Stuart, "The advanced theory of statistics. Distribution theory" , 3. Design and analysis , Griffin (1969) |

**How to Cite This Entry:**

Arcsine distribution.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Arcsine_distribution&oldid=18892