Muth distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Muth(κ) is used to indicate that the random variable X has the Muth distribution with parameter κ. A Muth random variable X with parameter κ has probability density function

f(x) = (e^{κx}− κ)e^{[− \frac{e^{κx}}{κ}+κx+ \frac{1}{κ}]} x > 0

for 0 < κ ≤ 1. The probability density function for three different values of κ is illustrated below.

PSfrag replacements

κ = 0.2
κ = 0.5
κ = 1.0
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − e^{[− \frac{e^{κx}}{κ}+κx+ \frac{1}{κ}]} x > 0.

The survivor function on the support of X is

S(x) = P(X ≥ x) = e^{[− \frac{e^{κx}}{κ}+κx+ \frac{1}{κ}]} x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = e^{κx}− κ x > 0.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = \frac{e^{κx}}{κ} − κx − \frac{1}{κ} x > 0.

The inverse distribution function, median, moment generating function and characteristic function of X are not mathematically tractable.

The population mean is

E[X] = 1.

APPL verification: The APPL statements

X := MuthRV(kappa); CDF(X); SF(X); HF(X); CHF(X); Mean(X);

verify the verify the cumulative distribution, survivor function, hazard function, cumulative hazard function, population mean.