玻尔兹曼(截断普朗克)分布¶
\BEGIN{eqnarray [}} p\left(k;N,\lambda\right) & = & \frac{{1-e^{{-\lambda}}}}{{1-e^{{-\lambda N}}}}\exp\left(-\lambda k\right)\quad k\in\left\{{ 0,1,\ldots,N-1\right\}} \\ F\left(x;N,\lambda\right) & = & \left\{{ \begin{{array}}{{cc}} 0 & x<0\\ \frac{{1-\exp\left[-\lambda\left(\left\lfloor x\right\rfloor +1\right)\right]}}{{1-\exp\left(-\lambda N\right)}} & 0\leq x\leq N-1\\ 1 & x\geq N-1\end{{array}}\right.\\ G\left(q,\lambda\right) & = & \left\lceil -\frac{{1}}{{\lambda}}\log\left[1-q\left(1-e^{{-\lambda N}}\right)\right]-1\right\rceil \end{{eqnarray] }
定义 \(z=e^{{-\lambda}}\)
\BEGIN{eqnarray [}} \mu & = & \frac{{z}}{{1-z}}-\frac{{Nz^{{N}}}}{{1-z^{{N}}}}\\ \mu_{{2}} & = & \frac{{z}}{{\left(1-z\right)^{{2}}}}-\frac{{N^{{2}}z^{{N}}}}{{\left(1-z^{{N}}\right)^{{2}}}}\\ \gamma_{{1}} & = & \frac{{z\left(1+z\right)\left(\frac{{1-z^{{N}}}}{{1-z}}\right)^{{3}}-N^{{3}}z^{{N}}\left(1+z^{{N}}\right)}}{{\left[z\left(\frac{{1-z^{{N}}}}{{1-z}}\right)^{{2}}-N^{{2}}z^{{N}}\right]^{{3/2}}}}\\ \gamma_{{2}} & = & \frac{{z\left(1+4z+z^{{2}}\right)\left(\frac{{1-z^{{N}}}}{{1-z}}\right)^{{4}}-N^{{4}}z^{{N}}\left(1+4z^{{N}}+z^{{2N}}\right)}}{{\left[z\left(\frac{{1-z^{{N}}}}{{1-z}}\right)^{{2}}-N^{{2}}z^{{N}}\right]^{{2}}}}\end{{eqnarray] }
\[M\left(t\right)=\frac{1-e^{N\left(t-\lambda\right)}}{1-e^{t-\lambda}}\frac{1-e^{-\lambda}}{1-e^{-\lambda N}}\]