幂函数分布¶
贝塔分布的一个特例 \(b=1\) 。有一个形状参数 \(a>0\) 和支持 \(x\in\left[0,1\right]\) 。
\BEGIN{eqnarray *}} f\left(x;a\right) & = & ax^{{a-1}}\\ F\left(x;a\right) & = & x^{{a}}\\ G\left(q;a\right) & = & q^{{1/a}}\\ \mu & = & \frac{{a}}{{a+1}}\\ \mu_{{2}} & = & \frac{{a\left(a+2\right)}}{{\left(a+1\right)^{{2}}}}\\ \gamma_{{1}} & = & 2\left(1-a\right)\sqrt{{\frac{{a+2}}{{a\left(a+3\right)}}}}\\ \gamma_{{2}} & = & \frac{{6\left(a^{{3}}-a^{{2}}-6a+2\right)}}{{a\left(a+3\right)\left(a+4\right)}}\\ m_{{d}} & = & 1\end{{eqnarray* }
\[H\Left [X\right] =1-\frac{1}{a}-\log\Left(a\Right)\]