Fall 2011 Strategic Practice 9: Section 2 (Order Statistics) - Question 3
Let $X_{1}, X_{2}, ..., X_{n}$ be i.i.d. r.v.s with CDF F and PDF f. Find the joint PDF of the order statistics $X_{(i)}$ and $X_{(j)}$ for $1\leq i\leq j\leq n$, by drawing and thinking about a picture.
Solution: To have X(i) be in a tiny interval around a and X(j) be in a tiny interval around b, where a < b, we need to have 1 of the Xk’s be almost exactly at a, another be almost exactly at b, i − 1 of them should be to the left of a, n − j should be to the right of b, and the remaining j −i−1 should be between a and b. See iTunes for more detail, including how we use the same idea used to find the coefficient in front of the Multinomial PMF.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."