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

f(x) = \frac{β}{α} x^{β−1} e^{−(1/α)x^{β}} x > 0.

The Weibull distribution is used in reliability and survival analysis to model the lifetime of an object, the lifetime of a organism, or a service time. The accelerated life and Cox proportional hazards model are identical when the baseline distribution is Weibull. The probability density function is plotted below for α = 1 and β = 1/2, 1, 2, 3.

PSfrag replacements

x
f(x)
β = 1
β = 2
β = 3
β = 1/2

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − e^{−(1/α)x^{β}} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = e^{−(1/α)x^{β}} x > 0.

The hazard function on the support of X is

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

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = \frac{1}{α} x^{β} x > 0.

The inverse distribution function of X is

F^{−1} (u) = [−αln(1 − u)] ^{1/β} 0 < u < 1.

The median of X is

(α ln2)^{1/β}.

The moment generating function of X is

M(t) = E [e^{tX}] = ∫_{0}^{∞} e^{tx}\frac{β}{α} x^{β−1} e^{−(1/α)x^{β}} dx t > 0.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = ∫_{0}^{∞} e^{itx}\frac{β}{α} x^{β−1} e^{−(1/α)x^{β}} dx t > 0.

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

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

E [( \frac{X − μ}{σ} ) ^{3}] = \{ \frac{2}{β}Γ( \frac{2}{β}) − [ \frac{1}{β}Γ( \frac{1}{β}) ]^{2}\} ^{−3/2} \{ \frac{3}{β}Γ( \frac{3}{β}) − \frac{6}{β^{2}}Γ( \frac{1}{β}) Γ( \frac{2}{β}) + 2[ \frac{1}{β}Γ( \frac{1}{β}) ]^{3}\}

E [( \frac{X − μ}{σ} ) ^{4}] = \{ \frac{2}{β}Γ( \frac{2}{β}) − [ \frac{1}{β}Γ( \frac{1}{β}) ]^{2}\} ^{−2} \{ \frac{4}{β}Γ( \frac{4}{β}) − \frac{12}{β^{2}}Γ( \frac{1}{β}) Γ( \frac{3}{β})

+\frac{12}{β^{3}} [Γ( \frac{1}{β}) ]^{2}Γ( \frac{2}{β}) − \frac{3}{β^{4}} [Γ( \frac{1}{β}) ]^{4}\} .

APPL verification: The APPL statements

X := WeibullRV(1 / alpha, beta); CDF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

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