Zeta distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Zeta(α) is used to indicate that the random variable X has the Zeta distribution with parameter α > 1. A Zeta random variable X with parameter α has probability mass function

f(x) = \frac{1}{x^{α} ∑_{i=1}^{∞} (1/i)^{α}} x = 0,1,2,...

for any α > 1. The probability mass function for α = 2 is illustrated below.

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The probability mass function can also be expressed as

f(x) = \frac{1}{x^{α}ζ(α)} x = 1,2,...,

where ζ(·) is the Riemann zeta function defined as

ζ(α) = ∑_{i=1}^{∞} (1/i)^{α}.

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{∑ _{i=1}^{x} (1/i)^{α}}{ζ(α)} x = 1,2,... .

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{∑ _{i=x}^{∞}(1/i)^{α}}{ζ(α)} x = 1,2,... .

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{1}{∑ _{i=x+1}^{∞} (1/i)^{α}} x = 1,2,... .

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = ln(ζ(α)) − ln( ∑_{i=x}^{∞}(1/i)^{α}) x = 1,2,... .

The inverse distribution function of X is mathematically intractable. The moment generating function of X is

M(t) = E [e^{tX}] = \frac{1}{ζ(α)} ∑_{x=1}^{∞} \frac{e^{tx}}{x^{α}} x = 1,2,... .

The characteristic function of X is

ϕ(t) = E [e^{itX}] = \frac{1}{ζ(α)} ∑_{x=1}^{∞} \frac{e^{itx}}{x^{α}} x = 1,2,... .

The population mean and variance of X are

E[X] = \frac{ζ(α − 1)}{ζ(α)} α > 2,

V [X] = \frac{ζ(α)ζ(α − 2) − ζ(α − 1)^{2}}{ζ(α)^{2}} α > 3.

APPL verification: The APPL statements

assume(alpha ¿ 1); X := [[x -¿ 1 / (x ˆ alpha * sum((1 / i ˆ alpha), i = 1 .. infinity))],

[0 .. infinity], [”Discrete”, ”PDF”]]; CDF(X); SF(X); HF(X); CHF(X); IDF(X); MGF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

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