Discrete Weibull distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Discrete Weibull(p, β) is used to indicate that the random variable X has the discrete Weibull distribution with real parameter p satisfying 0 < p < 1, and positive shape parameter β. A discrete Weibull random variable X with parameters p and β has probability mass function

f(x) = (1 − p)^{x^{β}}− (1 − p)^{(x+1)^{β}} x = 0,1,2,....

The probability mass function for three different parameter settings is illustrated below.

PSfrag replacements

x f(x)

p = 0.3 p = 0.6

β = 1 β = 2

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − (1 − p) ^{(x+1)^{β}} x = 0,1,2,....

The survivor function on the support of X is

S(x) = P(X ≥ x) = (1 − p) ^{x^{β}} x = 0,1,2,....

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = 1 − (1 − p) ^{(x+1)^{β}−x^{β}} x = 0,1,2,....

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −x^{β}ln(1 − p) x = 0,1,2,....

The inverse distribution function of X is mathematically intractable.

The moment generating function of X is

M(t) = E [e^{tX}] = ∑_{x=0}^{∞} ( (1 − p)^{x^{β}}− (1 − p)^{(x+1)^{β}} ) e^{tx}.

The population mean of X is

E [X]= ∑_{x=0}^{∞} ( (1 − p) ^{x^{β}}− (1 − p) ^{(x+1)^{β}}) x.

The variance, skewness, and kurtosis of X are mathematically intractable in the generalized case.

APPL verification: The APPL statements

X := [[x-¿(1-p)ˆ (xˆ b)-(1-p)ˆ ((x+1)ˆ b)], [0 .. infinity], [”Discrete”, ”PDF”]]; CDF(X); SF(X); CHF(X); MGF(X); Mean(X);

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