Arcsin distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ arcsinis used to indicate that the random variable X has the arcsin distribution. An arcsin random variable X has probability density function

f(x) = \frac{1}{π\sqrt{x(1 − x)}} 0 < x < 1.

The probability density function is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution on the support of X is

F (x) = P(X ≤ x) = \frac{π + 2arcsin(2x − 1)}{2π} 0 < x < 1.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{π − 2arcsin(2x − 1)}{2π} 0 < x < 1.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{2}{\sqrt{x(1 − x)}(π − 2arcsin(2x − 1))} 0 < x < 1.

The inverse distribution function of X is

F^{−1} (u) = \frac{1}{2} − \frac{1}{2} cos(πu) 0 < u < 1.

The cumulative hazard function, moment generating function, and characteristic function on the support of X are mathematically intractable.

The population mean, variance, skewness, and kurtosis of X are

E[X] = \frac{1}{2}  V [X] = \frac{1}{8}  E [( \frac{X − μ}{σ} ) ^{3}] = 0  E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3}{2}.

APPL verification: The APPL statements

X := [[x -¿ 1 / (Pi * sqrt(x * (1 - x)))], [0, 1], [”Continuous”, ”PDF”]]; CDF(X); SF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the cumulative distribution function, survivor function, hazard function, inverse distribution function, population mean, variance, skewness, kurtosis, and moment generating function.