Summary: AN EXAMPLE INVOLVING "ORDER STATISTICS"
PROF. GREG W. ANDERSON
This is a summary of an example considered from various angles in lecture on
October 6, 2011, worked up in the same style as the "rotten egg" example. I also
tack on the somewhat grundgy solution of problem 7 of §3.6 of our text.
Setup for the example
Suppose that two numbers are picked uniformly and independently from the unit
interval. Let X be the smaller number and let Y be the larger number.
The joint distribution of X and Y
The joint p.d.f. of X and Y . It can be shown that
fX,Y (x, y) = 2 for 0 < x < y < 1.
The "sacred rule" holds because the area of the triangle 0 < x < y < 1 is 1/2.
Calculation of a typical probability. We have
Pr(X 1/2 and Y 1/2) =
2 dx dy = 1/4.
Another way to compute this probability is to see that it equals the area of the