Let X1 X2 X3 be iid random variables with density fx Let Y1

Let X1, X2, X3 be iid random variables with density f(x). Let Y1 denote the smallest of {x1,x2,x3}; Y2 denote the second smallest of {x1,x2,x3}; and Y3, the largest of {x1,x2,x3}. Let X = (X1,X2,X3) and Y = (Y1,Y2,Y3) denote the corresponding random vectors.

1. Find the joint density of Y. Hints:

• Introduce some notation for sorted triplets, say S = {(x1, x2, x3) : x1 < x2 < x3}.

• Note that the event (Y A) can be decomposed into disjoint sets by considering all six possible one-to-one mapping : {1, 2, 3} {1, 2, 3}: (Y A) = ((X(1), X(2), X(3)) S A).

2. If a sample of three iid uniform random variables on (0, 1) is observed, what it the probability that the median (second smallest) is between 1/5 and 4/5?

Solution