カイ二乗分布の期待値と分散

カイ二乗分布の期待値と分散

カイ二乗分布の期待値(一次モーメント)

$$\begin{array}{rcl}\mathrm{E}\left[x\right]& =& {\int }_0^{\mathrm{\infty }}x{\chi }^2(x)\mathrm{d}x\\ & =& {\int }_0^{\mathrm{\infty }}x\frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}-1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}x{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}-1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}}{2}^{\frac{n}{2}}{\left(\frac{1}{2}\right)}^{\frac{n}{2}}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{2}^{\frac{n}{2}}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}}{\left(\frac{1}{2}\right)}^{\frac{n}{2}}\mathrm{d}x\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{\left(\frac{x}{2}\right)}^{\frac{n}{2}}\mathrm{d}x\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}}2\mathrm{d}t\cdots t=\frac{x}{2},\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{1}{2},\mathrm{d}x=2\mathrm{d}t\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}+1-1}\mathrm{d}t\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2\mathrm{\Gamma }(\frac{n}{2}+1)\cdots \mathrm{\Gamma }\left(s\right)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\\ & =& \frac{1}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2\frac{n}{2}\mathrm{\Gamma }\left(\frac{n}{2}\right)\cdots \mathrm{\Gamma }(s+1)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^s\mathrm{d}t=s\mathrm{\Gamma }(s)\\ & =& n\end{array}$$

カイ二乗分布の二次モーメント

$$\begin{array}{rcl}\mathrm{E}\left[x^2\right]& =& {\int }_0^{\mathrm{\infty }}{x}^2{\chi }^2(x)\mathrm{d}x\\ & =& {\int }_0^{\mathrm{\infty }}{x}^2\frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}-1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{x}^2{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}-1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}+1}{2}^{\frac{n}{2}+1}{\left(\frac{1}{2}\right)}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{2}^{\frac{n}{2}+1}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}+1}{\left(\frac{1}{2}\right)}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{2}^{\frac{n}{2}}2{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}+1}{\left(\frac{1}{2}\right)}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{1}{2^{\frac{n}{2}}\mathrm{\Gamma }\left(\frac{n}{2}\right)}{2}^{\frac{n}{2}}2{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{x}^{\frac{n}{2}+1}{\left(\frac{1}{2}\right)}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-\frac{x}{2}}{\left(\frac{x}{2}\right)}^{\frac{n}{2}+1}\mathrm{d}x\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}+1}2\mathrm{d}t\cdots t=\frac{x}{2},\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{1}{2},\mathrm{d}x=2\mathrm{d}t\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}+1}\mathrm{d}t\cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}+1+1-1}\mathrm{d}t\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2{\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{\frac{n}{2}+2-1}\mathrm{d}t\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2\mathrm{\Gamma }(\frac{n}{2}+2)\cdots \mathrm{\Gamma }\left(s\right)={\int }_0^{\mathrm{\infty }}{e}^{-t}{t}^{s-1}\mathrm{d}t\\ & =& \frac{2}{\mathrm{\Gamma }\left(\frac{n}{2}\right)}2(\frac{n}{2}+1)\frac{n}{2}\mathrm{\Gamma }\left(\frac{n}{2}\right)\cdots \mathrm{\Gamma }(s+2)=(s+1)\mathrm{\Gamma }(s+1)=(s+1)s\mathrm{\Gamma }(s)\\ & =& 2n(\frac{n}{2}+1)\\ & =& n(n+2)\end{array}$$

カイ二乗分布の分散(二次の中心モーメント)

$$\begin{array}{rcl}\mathrm{V}\left[x^2\right]& =& \mathrm{E}[(x-\mathrm{E}\left[x\right]{)}^2]\\ & =& \mathrm{E}\left[x^2\right]-\mathrm{E}{\left[x\right]}^2\\ & =& n(n+2)-n^2\\ & =& n^2+2n-n^2\\ & =& 2n\end{array}$$