標本平均の分布の歪度

標本平均($\bar{X}$)の分布の歪度

$$\begin{array}{rcl}{\displaystyle {\beta }_1(\bar{X})}& =& {\displaystyle \frac{E\left[{(\bar{X}-\mu )}^3\right]}{{\left(V{\left[\bar{X}\right]}^{\frac{1}{2}}\right)}^3}}\\ & =& {\displaystyle \frac{\frac{{\mu }_3}{n^2}}{{\left\{{\left(\frac{{\sigma }^2}{n}\right)}^{\frac{1}{2}}\right\}}^3}}& & {\displaystyle \dots E\left[{(\bar{X}-\mu )}^3\right]={\mu }_3\left(\bar{X}\right)=\frac{{\mu }_3}{n^2},V\left[\bar{X}\right]=\frac{{\sigma }^2}{n}}\\ & =& {\displaystyle \frac{\frac{{\mu }_3}{n^2}}{{\left(\frac{\sigma }{\sqrt{n}}\right)}^3}}\\ & =& {\displaystyle \frac{\frac{{\mu }_3}{n^2}}{\frac{{\sigma }^3}{n\sqrt{n}}}}\\ & =& {\displaystyle \frac{{\mu }_3}{n^2}\frac{n\sqrt{n}}{{\sigma }^3}}\\ & =& {\displaystyle \frac{{\mu }_3}{{\sigma }^3}\frac{1}{\sqrt{n}}}\\ & =& {\displaystyle \frac{{\beta }_1}{\sqrt{n}}}& & {\displaystyle \dots \frac{{\mu }_3}{{\sigma }^3}={\beta }_1}\end{array}$$ よって標本数$n$を増やすことで${\beta }_1\left(\bar{X}\right)$は$0$に近づいていく．