(統計)Chi-squared ($\chi^2$) distribution，卡方分布

 

Chi-squared ($\chi^2$) distribution，卡方分布

$\Gamma(\frac{r}{2},2)$

$f(x)=\left\{\begin{array}{l} \cfrac{1}{\Gamma(\cfrac{r}{2})2^{(\frac{r}{2})}}x^{(\frac{r}{2}-1)}e^{(\frac{-x}{2})}\ \ &,0<x<\infty\\0\ \ &,o.w\end{array}\right.$

$M(t)=(1-2t)^{\frac{-r}{2}}$

$Z_1,Z_2,\cdots,Z_n$ $\stackrel{ iid}{\sim}\ N(0,1)$ $\implies$ $X=\displaystyle\sum_{i=1}^{n}Z_i^2\ \sim \chi^2(n)$

$E(X)=n$，$Var(X)=2n$

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$\mu$已知，$\cfrac{nS^2}{\sigma^2}\ \sim \chi^2(n)$，$\mu$未知，
用$\overline{X}$取代，$\cfrac{(n-1)S^2}{\sigma^2}\ \sim \chi^2(n-1)$

證:

設   $\cfrac{X_i-\mu}{\sigma}\ \sim\ N(0,1)$ $\implies$ $\cfrac{(X_i-\mu)^2}{\sigma^2}\ \sim \chi^2(1)$ $\implies$ $\displaystyle\sum_{i=1}^{n}\cfrac{(X_i-\mu)^2}{\sigma^2}=\cfrac{(n-1)S^2}{\sigma^2}\ \sim\chi^2(n)$