標本平均の分布の歪度

標本平均(\(\overline{X}\))の分布の歪度

$$\begin{array}{rcl} \displaystyle \beta_1(\overline{X}) &=&\displaystyle \frac{E\left[\left(\overline{X}-\mu\right)^3\right]}{\left(V\left[\overline{X}\right]^{\frac{1}{2}}\right)^3}\\ &=&\displaystyle \frac{\frac{\mu_3}{n^2}}{\left\{ \left( \frac{\sigma^2}{n} \right) ^{\frac{1}{2}} \right\}^3} &&\displaystyle\,\dotso\,\href{https://shikitenkai.blogspot.com/2019/07/overlinexmu3.html}{E\left[\left(\overline{X}-\mu\right)^3\right]=\mu_3\left(\overline{X}\right)=\frac{\mu_3}{n^2}} ,\,\href{https://shikitenkai.blogspot.com/2019/06/specimen-random-variable_3.html}{V\left[\overline{X}\right]=\frac{\sigma^2}{n}}\\ &=&\displaystyle \frac{\frac{\mu_3}{n^2}}{\left( \frac{\sigma}{\sqrt{n}} \right)^3}\\ &=&\displaystyle \frac{\frac{\mu_3}{n^2}}{ \frac{\sigma^3}{n\sqrt{n}} }\\ &=&\displaystyle \frac{\mu_3}{n^2} \frac{n\sqrt{n}}{\sigma^3}\\ &=&\displaystyle \frac{\mu_3}{\sigma^3} \frac{1}{\sqrt{n}}\\ &=&\displaystyle \frac{\beta_1}{\sqrt{n}} &&\displaystyle\,\dotso\,\href{https://shikitenkai.blogspot.com/2019/07/mu-sigma2beta1.html}{\frac{\mu_3}{\sigma^3}=\beta_1}\\ \end{array}$$ よって標本数\(n\)を増やすことで\(\beta_1\left(\overline{X}\right)\)は\(0\)に近づいていく．