Zipf(Zeta)分布¶
随机变量具有带参数的Zeta分布(也称为Zipf分布 \(\alpha>1\) 如果它的概率质量函数由下式给出
\BEGIN{eqnarray *}} p\left(k;\alpha\right) & = & \frac{{1}}{{\zeta\left(\alpha\right)k^{{\alpha}}}}\quad k\geq1\end{{eqnarray* }
哪里
\[\zeta\left(\alpha\right)=\sum_{n=1}^{\infty}\frac{1}{n^{\alpha}}\]
是Riemann Zeta函数。此发行版的其他功能包括
\BEGIN{eqnarray [}} F\left(x;\alpha\right) & = & \frac{{1}}{{\zeta\left(\alpha\right)}}\sum_{{k=1}}^{{\left\lfloor x\right\rfloor }}\frac{{1}}{{k^{{\alpha}}}}\\ \mu & = & \frac{{\zeta_{{1}}}}{{\zeta_{{0}}}}\quad\alpha>2\\ \mu_{{2}} & = & \frac{{\zeta_{{2}}\zeta_{{0}}-\zeta_{{1}}^{{2}}}}{{\zeta_{{0}}^{{2}}}}\quad\alpha>3\\ \gamma_{{1}} & = & \frac{{\zeta_{{3}}\zeta_{{0}}^{{2}}-3\zeta_{{0}}\zeta_{{1}}\zeta_{{2}}+2\zeta_{{1}}^{{3}}}}{{\left[\zeta_{{2}}\zeta_{{0}}-\zeta_{{1}}^{{2}}\right]^{{3/2}}}}\quad\alpha>4\\ \gamma_{{2}} & = & \frac{{\zeta_{{4}}\zeta_{{0}}^{{3}}-4\zeta_{{3}}\zeta_{{1}}\zeta_{{0}}^{{2}}+12\zeta_{{2}}\zeta_{{1}}^{{2}}\zeta_{{0}}-6\zeta_{{1}}^{{4}}-3\zeta_{{2}}^{{2}}\zeta_{{0}}^{{2}}}}{{\left(\zeta_{{2}}\zeta_{{0}}-\zeta_{{1}}^{{2}}\right)^{{2}}}}.\end{{eqnarray] }
\BEGIN{eqnarray *}} M\left(t\right) & = & \frac{{\textrm{{Li}}_{{\alpha}}\left(e^{{t}}\right)}}{{\zeta\left(\alpha\right)}}\end{{eqnarray* }
哪里 \(\zeta_{{i}}=\zeta\left(\alpha-i\right)\) 和 \(\textrm{{Li}}_{{n}}\left(z\right)\) 是不是 \(n^{{\textrm{{th}}}}\) 的多对数函数 \(z\) 定义为
\[\textrm{Li}_{n}\left(z\right)\equiv\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\]
\[\mu_{n}^{\prime}=\left.M^{\left(n\right)}\left(t\right)\right |_{{t=0}}=\left.\frac{{\textrm{{Li}}_{{\alpha-n}}\left(e^{{t}}\right)}}{{\zeta\left(a\right)}}\right|_ {t=0}=\frac{\zeta\left(\alpha-n\right)}{\zeta\left(\alpha\right)}\]
实施： scipy.stats.zipf