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)

$$ \begin{eqnarray} \mathrm{E}\left[x\right]&=&\int_0^\infty x \chi^2(x) \mathrm{d}x \\&=&\int_0^\infty x \frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}e^{-\frac{x}{2}}x^{\frac{n}{2}-1} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty x e^{-\frac{x}{2}}x^{\frac{n}{2}-1} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}} \color{red}{2^{\frac{n}{2}}\left(\frac{1}{2}\right)^{\frac{n}{2}}} \color{black}{\mathrm{d}x} \\&=&\frac{1}{\color{red}{2^{\frac{n}{2}}}\color{black}{\Gamma\left(\frac{n}{2}\right)}}\color{red}{2^{\frac{n}{2}}}\color{black}{\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}} \left(\frac{1}{2}\right)^{\frac{n}{2}}\mathrm{d}x} \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}\left(\frac{x}{2}\right)^{\frac{n}{2}}\mathrm{d}x \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-t}t^{\frac{n}{2}}\;2\mathrm{d}t \;\cdots\;t=\frac{x}{2},\frac{\mathrm{d}t}{\mathrm{d}x}=\frac{1}{2},\mathrm{d}x=2\mathrm{d}t \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}2\int_0^\infty e^{-t}t^{\frac{n}{2}}\;\mathrm{d}t \;\cdots\;\int cf(x) \mathrm{d}x=c\int f(x) \mathrm{d}x \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}2\int_0^\infty e^{-t}t^{\frac{n}{2}\color{red}{+1-1}}\;\mathrm{d}t \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}2\Gamma\left(\frac{n}{2}+1\right) \;\cdots\;\Gamma\left(s\right)=\int_0^\infty e^{-t}t^{s-1}\mathrm{d}t \\&=&\frac{1}{\Gamma\left(\frac{n}{2}\right)}2\frac{n}{2}\Gamma\left(\frac{n}{2}\right) \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/s1ss.html}{\Gamma(s+1)=\int_0^\infty e^{-t}t^{s}\;\mathrm{d}t=s\Gamma(s)} \\&=&n \end{eqnarray} $$

カイ二乗分布の二次モーメント

$$ \begin{eqnarray} \mathrm{E}\left[x^2\right]&=&\int_0^\infty x^2 \chi^2(x) \mathrm{d}x \\&=&\int_0^\infty x^2 \frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}e^{-\frac{x}{2}}x^{\frac{n}{2}-1} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty x^2 e^{-\frac{x}{2}}x^{\frac{n}{2}-1} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}+1} \mathrm{d}x \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}+1} \color{red}{2^{\frac{n}{2}+1} \left(\frac{1}{2}\right)^{\frac{n}{2}+1} } \color{black}{\mathrm{d}x} \\&=&\frac{1}{2^{\frac{n}{2}}\Gamma\left(\frac{n}{2}\right)}2^{\frac{n}{2}+1}\int_0^\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}}\Gamma\left(\frac{n}{2}\right)}2^{\frac{n}{2}}2\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}+1} \left(\frac{1}{2}\right)^{\frac{n}{2}+1}\mathrm{d}x \\&=&\frac{1}{\color{red}{2^{\frac{n}{2}}}\color{black}{\Gamma\left(\frac{n}{2}\right)}}\color{red}{2^{\frac{n}{2}}}\color{black}{2\int_0^\infty e^{-\frac{x}{2}}x^{\frac{n}{2}+1} \left(\frac{1}{2}\right)^{\frac{n}{2}+1}\mathrm{d}x} \\&=&\frac{2}{\Gamma\left(\frac{n}{2}\right)}\int_0^\infty e^{-\frac{x}{2}}\left(\frac{x}{2}\right)^{\frac{n}{2}+1}\mathrm{d}x \\&=&\frac{2}{\Gamma\left(\frac{n}{2}\right)}\int_0^\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}{\Gamma\left(\frac{n}{2}\right)}2\int_0^\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}{\Gamma\left(\frac{n}{2}\right)}2\int_0^\infty e^{-t}t^{\frac{n}{2}+1\color{red}{+1-1}}\;\mathrm{d}t \\&=&\frac{2}{\Gamma\left(\frac{n}{2}\right)}2\int_0^\infty e^{-t}t^{\frac{n}{2}+2-1}\;\mathrm{d}t \\&=&\frac{2}{\Gamma\left(\frac{n}{2}\right)}2\Gamma\left(\frac{n}{2}+2\right) \;\cdots\;\Gamma\left(s\right)=\int_0^\infty e^{-t}t^{s-1}\mathrm{d}t \\&=&\frac{2}{\Gamma\left(\frac{n}{2}\right)}2\left(\frac{n}{2}+1\right)\frac{n}{2}\Gamma\left(\frac{n}{2}\right) \;\cdots\;\href{https://shikitenkai.blogspot.com/2020/08/s1ss.html}{\Gamma\left(s+2\right)=(s+1)\Gamma(s+1)=(s+1)s\Gamma(s)} \\&=&2n\left(\frac{n}{2}+1\right) \\&=&n(n+2) \end{eqnarray} $$

カイ二乗分布の分散(二次の中心モーメント)

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