帕累托分布¶
一个形状参数 \(b>0\) 和支持 \(x\geq1\) 。标准格式为
\BEGIN{eqnarray *}} f\left(x;b\right) & = & \frac{{b}}{{x^{{b+1}}}}\\ F\left(x;b\right) & = & 1-\frac{{1}}{{x^{{b}}}}\\ G\left(q;b\right) & = & \left(1-q\right)^{{-1/b}}\end{{eqnarray* }
\BEGIN{eqnarray *}} \mu & = & \frac{{b}}{{b-1}}\quad b>1\\ \mu_{{2}} & = & \frac{{b}}{{\left(b-2\right)\left(b-1\right)^{{2}}}}\quad b>2\\ \gamma_{{1}} & = & \frac{{2\left(b+1\right)\sqrt{{b-2}}}}{{\left(b-3\right)\sqrt{{b}}}}\quad b>3\\ \gamma_{{2}} & = & \frac{{6\left(b^{{3}}+b^{{2}}-6b-2\right)}}{{b\left(b^{{2}}-7b+12\right)}}\quad b>4\end{{eqnarray* }
\[h\left(X\right)=\frac{1}{c}+1-\log\left(c\right)\]