Order Statistics

Definition

• For Random Variable $X_1,\dots ,X_n$
• The $k-th$ order statistic $(X_{(k)})$ is the $k$-th smallest variable
  • $X_{(1)}=min(X_1,\dots ,X_n)$
  • $X_{(1)}\le X_{(2)}\le \cdots \le X_{(n-1)}\le X_{(n)}$
  • $X_{(n)}=max(X_1,\dots ,X_n)$

Easiest to find when a CDF is known, or the RV are Independent and Identically Distributed Random Variables