Order Statistic

Motivation. When you have iid random variables $X_1,\dots ,X_n$ with a known pdf, you sometimes want the pdf of the minimum, maximum or k-th smallest/biggest of the realizations.

def. $k$-th order statistic $X_{(k)}$is the $k$-th smallest of iid rv $X_1,\dots ,X_n$. Equivalently, $X_{(1)}$ is the smallest, and $X_{(n)}$ is the largest.

thm. pdf of order statistic. For iid rv $X_1,\dots ,X_n$:

$$\begin{array}{rl}P(X_{(k)}\in [x,x+ϵ])& =P(\text{one of the X’s}\in [x,x+ϵ]\text{ and exactly }k-1\text{ of the others }<x)\\ & =\sum _{i=1}^{n}P(X_i\in [x,x+ϵ]\text{ and exactly }k-1\text{ of the others }<x)\\ & =nP(X_1\in [x,x+ϵ]\text{ and exactly }k-1\text{ of the others }<x)\\ & =nP(X_1\in [x,x+ϵ])P(\text{exactly }k-1\text{ of the others }<x)\\ & =nP(X_1\in [x,x+ϵ])\left(\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)P(X<x{)}^{k-1}P(X>x{)}^{n-k}\right)\end{array}$$

And the pdf is:

$$f_{(k)}(x)=nf(x)\left(\genfrac{}{}{0ex}{}{n-1}{k-1}\right)F(x{)}^{k-1}(1-F(x){)}^{n-k}$$