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

f(x) = −\frac{(1 − c) ^{x}}{xlnc} x = 1,2,...

for any 0 < c < 1. The probability mass functions for two different values of c are illustrated below.

PSfrag replacements

f(x)

x c = 0.1 c = 0.5

The cumulative distribution, survivor function, hazard function, cumulative hazard function, and inverse distribution function on the support of X are mathematically intractable.

The moment generating function of X is

M(t) = E [e^{tX}] = \frac{ln(1 − e^{t}+ e^{t}c)}{lnc} − ∞ < t < ∞.

The characteristic function of X is

ϕ(t) = \frac{ln(1 − e^{it}+ e^{it}c)}{lnc} − ∞ < t < ∞.

The population mean and variance of X are

E[X] = \frac{c − 1}{clnc} V [X] = \frac{(c − 1) (ln(c)+ 1 − c)}{(lnc)^{2}c^{2}}

The skewness and kurtosis can be determined using the APPL code below.

APPL verification: The APPL statements

assume(0 ¡ c ¡ 1); X := [[x -¿ -(1 - c) ˆ x / (x * log(c))],[1 .. infinity], [”Discrete”, ”PDF”]]; Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

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