Logistic distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ logistic(λ,κ) is used to indicate that the random variable X has the logistic distribution with parameters λ and κ. A logistic random variable X with positive scale parameter λ and positive shape parameter κ has probability density function

f(x) = \frac{λ^{κ}κe^{κx}}{(1 + (λe^{x})^{κ})^{2}} − ∞ < x < ∞.

The probability density function with three different parameter settings is illustrated below.

PSfrag replacements λ = .05, κ = 3 λ = .001, κ = .75 λ = .005, κ = 1.5

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{λ^{κ}e^{κx}}{1 + λ^{κ}e^{x}} − ∞ < x < ∞.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{1 + λ^{κ}e^{x}− λ^{κ}e^{xκ}}{1 + λ^{κ}e^{x}} − ∞ < x < ∞.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{κe^{κx}(λ^{κ} + λ^{2κ}e^{x})}{(1 + (λe^{x})^{κ})^{2}(1 + λ^{κ}e^{x}− λ^{κ}e^{xκ})} − ∞ < x < ∞.

The inverse distribution function of X is

F^{−1} (u) = −\frac{1}{κ} ln( \frac{u}{1 − u} ) − lnλ 0 < u < 1.

The moment generating function of X is

M(t) = E [e^{tX}] = ∫_{−∞}^{∞} \frac{λ^{κ}κe^{x(t+κ)}}{(1 + λ^{κ}e^{xκ})^{2}} dx.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = ∫_{−∞}^{∞} \frac{λ^{κ}κe^{x(it+κ)}}{(1 + λ^{κ}e^{xκ})^{2}} dx.

The population mean and variance are

E[X] = −lnλ V [X] = \frac{π^{2}}{3κ^{2}}.

The population skewness and kurtosis of X are more complicated expressions.

APPL verification: The APPL statements

X := LogisticRV(kappa, lambda); CDF(X); SF(X); HF(X); IDF(X); MGF(X); Mean(X); Variance(X);

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