Standard power distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ power(1,β) is used to indicate that the random variable X has the standard power distribution with shape parameter β > 0. A standard power random variable X with parameter β has probability density function

f(x) = βx^{β−1} 0 < x < 1.

The probability density function with two different values of β is illustrated below.

PSfrag replacements β = 0.5

β = 3
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = x^{β} 0 < x < 1.

The survivor function on the support of X is

S(x) = P(X ≥ x) = 1 − x^{β} 0 < x < 1.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{βx^{β−1}}{1 − x^{β}} 0 < x < 1.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −ln(1 − x^{β}) 0 < x < 1.

The inverse distribution function of X is

F^{−1} (u) = u^{1/β} 0 < u < 1.

The median of X is

( \frac{1}{2}) ^{1/β}.

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

E[X] = \frac{β}{β + 1} V [X] = \frac{β}{(β + 2)(β + 1)^{2}}

E [( \frac{X − μ}{σ} ) ^{3}] = \frac{2(1 − β)\sqrt{β + 2}}{(β + 3)\sqrt{β}} E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(3β^{2}− β + 2) (β + 2)}{β(β + 3)(β + 4)} .

APPL verification: The APPL statements

assume(beta ¿ 0); X := [[x -¿ beta * x ˆ (beta - 1)], [0, 1], [”Continuous”, ”PDF”]]; Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the population mean, variance, skewness, kurtosis, and moment generating function.