Throw two darts at a circular dartboard of radius 1. Suppose the points where the darts hit are independent and are uniformly distributed on the face of the dartboard.
Let X be the distance to the center for the first throw and let Y be the distance to the center for the second throw. Recall that the density of X is f(x) = 2x for 0 < x < 1, and otherwise f(x) equals 0.
Y has the same density as X, since X and Y behave the same way.
The "joint density" of X and Y is a surface over plane for which probability for the random point (X, Y) equals volume under that surface.
In this case, the joint density is f(x, y) = (2x)(2y) = 4xy for all (x,y) points with 0 < x < 1 and 0 < y < 1, and equals 0 otherwise.
So, for any region A in the (x, y) plane, the probability that the random point (X, Y) lands in region A is the double integral of f(x, y) = 4xy over the intersection of region A with the unit square.
Let T = X + Y.
(a). Find P (T < 1).
(b). Find P (2Y < X).
(c). Let U = min (X, Y) and V = max (X, Y). Find the joint density f(u, v) of U and V.
(d). Are U and V independent? (See formula (1) on page 350 for the general definition of independence for random variables.)
(e). Find P (2U < V).