Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 12 / c / Solution

For a small rectangle of side lengths $ds,dt$, independence makes its probability approximately $f_S(s)ds{f}_T(t)dt$. Thus the joint probability density function is the productAdding the probabilities of disjoint small rectangles covering a region of integration, and taking their sizes to zero, gives the double integral of this product over that region. Without independence, the two marginal distributions alone would not determine the joint probability density function.