Some Properties of the Arcsine Distribution

Abstract

Three characterizations of the arcsine distribution are presented. A symmetric random variable X has the arcsine distribution if and only if X 2 and (1 + X ) /2 are identically distributed. Also, a symmetric random variable has an arcsine distribution if and only if X 2 and 1 – X 2 are identically distributed and X and 2 X (1 – X 2 ) are identically distributed. A related result dealing with two i.i.d. random variables is also presented. The fact that the sine of a uniform random variable has the arcsine distribution is used to motivate these characterizations. One of the characterizations is used to quantify the heavy-tailed nature of the arcsine distribution.