 Solution

ID: 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

d s, d t

, independence makes its probability approximately

f_{S} (s) d s f_{T} (t) d t

f_{S, T} (s, t) = \frac{1}{s ˉ t ˉ} e^{- s / \overset{s}{ˉ} - t / \overset{ˉ}{t}}, s, t \geq 0.

Adding 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.

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