# Zipfian Distribution

A random variable has the Zipfian distribution with parameters \(s \ge 0\) and \(N \in \{1, 2, 3, \dots\}\) if its probability mass function is given by

\begin{eqnarray*} p\left(k; s, N \right) & = & \frac{1}{H_{N, s}k^{s}}\quad k \in \{1, 2, \dots, n-1, n\} \end{eqnarray*}

where

\[H_{N, s}=\sum_{n=1}^{N}\frac{1}{n^{s}}\]

is the \(N\)th generalized harmonic number of order \(s\). Other functions of this distribution are

\begin{eqnarray*} F\left(x; s, N\right) & = & \frac{H_{k, s}}{H_{N, s}}, \\ \mu & = & \frac{H_{N, s-1}}{H_{N, s}},\\ \mu_{2} & = & \frac{H_{N, s-2}}{H_{N, s}} - \frac{H^2_{N, s-1}}{H^2_{N, s}},\\ \gamma_1 & = & \frac{\frac{H_{N, s-3}}{H_{N, s}} - 3 \frac{H_{N, s-1}H_{N, s-2}}{H_{N, s}^2} + 2\frac{H_{N, s-1}^3}{H_{N, s}^3}}{\left(\frac{H_{N, s-2}H_{N, s}- H_{N, s-1}^2}{H_{N, s}^2}\right)^{\frac{3}{2}}}, \mbox{and}\\ \gamma_2 & = & \frac{H_{N, s}^3 H_{N, s-4} - 4 H_{N, s}^2 H_{N, s-1} H_{N, s-3} + 6 H_{N, s} H_{N, s-1}^2 H_{N, s-2} - 3 H_{N, s-1}^4}{\left(H_{N, s-2} H_{N, s} - H_{N, s-1}^2 \right)^2}. \end{eqnarray*}

## References

- “Zipf’s Law”, Wikipedia, https://en.wikipedia.org/wiki/Zipf%27s_law

- Larry Leemis, “Zipf Distribution”, Univariate Distribution Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf

Implementation: `scipy.stats.zipfian`