 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/7/b/ii/solution

Split the unit square into

v \leq u

and

v > u

. The maximum is

u

on the first triangle and

v

on the second, so

E [U max {U, V}] = \int_{0}^{1} [\int_{0}^{u} u^{2} d v + \int_{u}^{1} uv d v] d u = \int_{0}^{1} [u^{3} + \frac{u}{2} (1 - u^{2})] d u = \frac{1}{8} + \frac{1}{4} = \frac{3}{8} .

The shared boundary has zero probability, so it is not counted twice in this joint density calculation.

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