Joint pdf of Order Statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with
  • pdf $f(x)$
  • Support $(a,b)$
• $Y_1<\cdots <Y_n$ : $n$ order statistics of $X_1,\dots ,X_n$

Then the joint pdf of $Y_1,\dots ,Y_n$ is

$$g(y_1,\dots ,y_n)=\{\begin{array}{ll}n!f(y_1)\dots f(y_n)& a<y_1<\cdots <y_n<b\\ 0& \text{elsewhere}\end{array}$$