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

f(x) = \frac{κλ^{κ}}{x^{κ+1}} x > λ.

The Pareto distribution has traditionally been used to model the distribution of income, where λ is a minimum wage and κ models the distribution of the income. The Pareto distribution can also be used to model the lifetime of an object with a warranty period λ or the duration of a strike with minimum duration λ. The probability density function with λ = 1 and two different values of κ is illustrated below.

PSfrag replacements

κ = 1
κ = 4
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = 1 − ( \frac{λ}{x}) ^{κ} x > λ.

The survivor function on the support of X is

S(x) = P(X ≥ x) = ( \frac{λ}{x}) ^{κ} x > λ.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{κ}{x} x > λ.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = −κ(lnλ − lnx) x > λ.

The inverse distribution function of X is

F^{−1} (u) = λ(1 − u)^{−1/κ} 0 < u < 1.

The median of X is

λ2^{1/κ}.

The moment generating function of X is

M(t) = E [e^{tX}] = κ(−λt)^{κ}Γ(−κ,−λt) − ∞ < t < ∞.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = κ(−λit)^{κ}Γ(−κ,−λit) − ∞ < t < ∞.

The population mean, variance, skewness, and kurtosis of X are

E[X] = \frac{κλ}{κ − 1} for κ > 1 V [X] = \frac{κλ^{2}}{(κ − 1)^{2}(κ − 2)} for κ > 2

E [( \frac{X − μ}{σ} ) ^{3}] = 2 \frac{κ + 1}{κ − 3} \sqrt{\frac{κ − 2}{κ}} forκ > 3 E [( \frac{X − μ}{σ} ) ^{4}] = \frac{3(κ − 2)(3κ^{3}+ κ + 2)}{κ(κ − 3)(κ − 4)} forκ > 4.

APPL verification: The APPL statements

X := ParetoRV(lambda, kappa); CDF(X); SF(X); HF(X); CHF(X): IDF(X); assume(kappa ¿ 1); Mean(X); assume(kappa ¿ 2); Variance(X); assume(kappa ¿ 3); Skewness(X); assume(kappa ¿ 4); Kurtosis(X); MGF(X);

verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, and inverse distribution function. Some of the population moments are expressed as limits.