cdf of Order Statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf $f(x)$
• $Y_1,\dots ,Y_n$ : Order statistics of $X_1,\dots ,X_n$

Then cdf of $Y_1,\dots ,Y_n$ is given by

• $Y_1$¹

$$F_{Y_1}(x)=1-[1-F_X(x){]}^n$$

• $Y_k$²

$$F_{Y_k}(x)=\sum _{j=k}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)[F_X(x){]}^j[1-F_X(x){]}^{n-j}$$

• $Y_n$¹

$$F_{Y_n}(x)=[F_X(x){]}^n$$

Example

Let $Y_1<Y_2<Y_3<Y_4$ be the order statistics of a random sample of size $4$ from the distribution having pdf $f(x)=e^{-x},0<x<\infty$, zero elsewhere.

Find $P(3\le Y_4)$.

$$\begin{array}{rl}P(3\le Y_4)& =1-P(Y_4\le 3)\\ & =1-F_{Y_4}(3)\\ & =1-[F_X(3){]}^4\\ & =1-{\left[{\int }_0^3{e}^{-x}dx\right]}^4\\ & =\boxed{0.1847}\end{array}$$

Footnotes

1. https://www.colorado.edu/amath/sites/default/files/attached-files/order_stats.pdf ↩ ↩²

2. https://en.wikipedia.org/wiki/Order_statistic#Cumulative_distribution_function_of_order_statistics (Taken in 2025-06-24) ↩