齐夫分布#
如果随机变量的概率质量函数由以下公式给出，则该随机变量具有参数为 \(s \ge 0\) 和 \(N \in \{1, 2, 3, \dots\}\) 的齐夫分布：
\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*}
其中
\[H_{N, s}=\sum_{n=1}^{N}\frac{1}{n^{s}}\]
是 \(s\) 阶的第 \(N\) 个广义调和数。该分布的其他函数为
\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*}
参考文献#
“齐夫定律”，维基百科，https://en.wikipedia.org/wiki/Zipf%27s_law
Larry Leemis，“齐夫分布”，单变量分布关系。 http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf