命题 6.3.5 (正态分布的顺序统计量)

设总体 $X$ 的分布函数为 $F_X$, 分布密度函数为 $f_X$, 则对 $k=1,2,\cdots ,n$

$$f_{X_{\left(k\right)}}\left(x\right)=\frac{n!}{\left(n-k\right)!\left(k-1\right)!}{\left[F_X\left(x\right)\right]}^{k-1}{\left[1-F_X\left(x\right)\right]}^{n-k}{f}_X\left(x\right).\text{\hspace{1em}(6.3.17)}$$

特别地,

$$f_{X_{\left(1\right)}}\left(x\right)=n{\left[1-F_X\left(x\right)\right]}^{n-1}{f}_X\left(x\right),\text{\hspace{1em}(6.3.18)}$$ $$f_{X_{\left(n\right)}}\left(x\right)=n{\left[F_X\left(x\right)\right]}^{n-1}{f}_X\left(x\right).\text{\hspace{1em}(6.3.19)}$$