对数(对数系列、系列)分布¶
带参数的对数分布 \(p\) 具有概率质量函数，其项与泰勒级数展开成正比 \(\log\left(1-p\right)\)
\BEGIN{eqnarray *}} p\left(k;p\right) & = & -\frac{{p^{{k}}}}{{k\log\left(1-p\right)}}\quad k\geq1\\ F\left(x;p\right) & = & -\frac{{1}}{{\log\left(1-p\right)}}\sum_{{k=1}}^{{\left\lfloor x\right\rfloor }}\frac{{p^{{k}}}}{{k}}=1+\frac{{p^{{1+\left\lfloor x\right\rfloor }}\Phi\left(p,1,1+\left\lfloor x\right\rfloor \right)}}{{\log\left(1-p\right)}}\end{{eqnarray* }
哪里
\[\Phi\Left(z，s，a\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{\left(a+k\right)^{s}}\]
是勒奇的超凡脱俗之作。还定义了 \(r=\log\left(1-p\right)\)
\BEGIN{eqnarray [}} \mu & = & -\frac{{p}}{{\left(1-p\right)r}}\\ \mu_{{2}} & = & -\frac{{p\left[p+r\right]}}{{\left(1-p\right)^{{2}}r^{{2}}}}\\ \gamma_{{1}} & = & -\frac{{2p^{{2}}+3pr+\left(1+p\right)r^{{2}}}}{{r\left(p+r\right)\sqrt{{-p\left(p+r\right)}}}}r\\ \gamma_{{2}} & = & -\frac{{6p^{{3}}+12p^{{2}}r+p\left(4p+7\right)r^{{2}}+\left(p^{{2}}+4p+1\right)r^{{3}}}}{{p\left(p+r\right)^{{2}}}}.\end{{eqnarray] }
\BEGIN{eqnarray *}} M\left(t\right) & = & -\frac{{1}}{{\log\left(1-p\right)}}\sum_{{k=1}}^{{\infty}}\frac{{e^{{tk}}p^{{k}}}}{{k}}\\ & = & \frac{{\log\left(1-pe^{{t}}\right)}}{{\log\left(1-p\right)}}\end{{eqnarray* }
因此，
\[\mu_{n}^{\prime}=\left.M^{\left(n\right)}\left(t\right)\right |_{{t=0}}=\left.\frac{{\textrm{{Li}}_{{1-n}}\left(pe^{{t}}\right)}}{{\log\left(1-p\right)}}\right|_ {t=0}=-\frac{\textrm{Li}_{1-n}\left(p\right)}{\log\left(1-p\right)}.\]