Ex:           Find for joint probability density function f(x, y) = fX(x) fY(y) where fX(x) is a uniform distribution on the interval [0, 1] and fY(y) is a triangular distribution on the interval [0,2].

Sol'n:      The illustration, below, shows the 3-D shape of f(x, y) and the volume of f(x, y) that lies over the region and is equal to .

The probability we are seeking is the volume of the small wedge. We may compute it by simple geometry as half the volume of a cube that measures 1/2 on each side:

By restricting the limits of integration to the region of the xy-plane where is nonzero, we have an expression for the probability we seek:

or

or

or