 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/6/a/ii/solution

Obtain each marginal density by integrating the joint probability density function over the other variable:

f_{U} (u) = \int_{0}^{1} u (1 + 2 v) d v = 2 u, 0 \leq u \leq 1,

f_{V} (v) = \int_{0}^{1} u (1 + 2 v) d u = \frac{1}{2} + v, 0 \leq v \leq 1.

Each marginal density is zero off its displayed interval and has total integral one.

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