Power distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ power(α, β) is used to indicate that the random variable X has the power distribution with parameters α and β. A power random variable X with positive scale parameter α and positive shape parameter β has probability density function

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

The probability density function with α = 1 and 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) = \frac{x^{β}}{α^{β}} 0 < x < α.

The survivor function on the support of X is

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

The hazard function on the support of X is

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

The cumulative hazard function on the support of X is

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

The inverse distribution function of X is

F^{−1} (u) = αu^{1/β} 0 < u < 1.

The median of X is

α( \frac{1}{2}) ^{1/β}

The moment generating function of X is

M(t) = E [e^{tX}] = −\frac{βΓ(β,−tα) − Γ(1 + β)}{(−t)^{β}α^{β}} − ∞ < t < ∞.

The characteristic function of X is mathematically intractable. The population mean, variance, skewness, and kurtosis of X are

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

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

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

APPL verification: The APPL statements

assume(alpha ¿ 0); assume(beta ¿ 0); X := [[x -¿ beta * x ˆ (beta - 1) / alpha ˆ beta], [0, alpha],

[”Continuous”, ”PDF”]]; CDF(X); SF(X); HF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

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