Ex:           For the following joint probability density function, plot the area on the x,y-plane where F(x, y) = 1. (In other words, plot the footprint of the area where F(x, y) = 1.)

       

Sol'n:      f(x, y) is a cylinder of height 1/π, centered on the origin:

F(x, y) equals the volume of f(x, y) to the left of x and in front of, (i.e., less than), y. The illustration, above, shows two walls at x and y. F(x, y) = 1 for the x and y shown, since all the volume of f(x, y) is left of x and in front of y. The region of the where F(x, y) = 1 becomes apparent in a top view:

All of the volume of f(x, y) will be to the left of x for x > 1 and in front of y for y > 1. Thus, the region where F(x, y) = 1 is x > 1 and y > 1. Note that both x and y conditions must be true.