Fall 2011 Strategic Practice 9: Section 2 (Order Statistics) - Question 1
Let $U_{1}, ..., U_{n}$ be i.i.d. Unif(0, 1). Find the PDF of the jth order statistic $U_{(j)}$ (including the normalizing constant), and its mean and variance.
Solution: U(j) ~ Beta(j, n − j + 1). Mean u = j / n+1. Variance u(1-u) / n+2.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."