IDB distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ IDB(δ,κ,γ) is used to indicate that the random variable X has the IDB distribution with parameters δ, κ, and γ. An IDB random variable X with parameters δ, κ, and γ has probability density function

f(x) = \begin{matrix}((1 + κx) δx + γ) & e^{−1/(2δx^{2})} \\ (1 + κx) & γ/κ+1 \end{matrix} x > 0,

for all δ > 0, κ > 0, and γ ≥ 0. The probability density function with three parameter settings is illustrated below.

PSfrag replacements δ = 1, γ = 2,κ = 10 δ = 2, γ = 5,κ = 1 δ = 2, γ = 0,κ = 1

x
f(x)

The cumulative distribution, survivor, hazard, cumulative hazard, inverse distribution, moment generating, and characteristic functions on the support of X are mathematically intractable. The population mean, variance, skewness, and kurtosis of X are mathematically intractable.

APPL failure: The APPL statements

assume(delta¿0); assume(kappa¿0); assume(y¿=0); X := [[x -¿ ((1 + kappa * x) * delta * x + y) / ((1 + kappa * x) ˆ (y / kappa + 1))

* exp(-delta * x ˆ 2 / 2)], [0,infinity], [”Continuous”, ”PDF”]];

CDF(X); SF(X); HF(X); CHF(X); IDF(X);

fail to yield the cumulative distribution, survivor, hazard, cumulative hazard, and inverse distribution functions.