 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/m/m2/b/iii/solution

E X = 12 \int_{0}^{1} x^{2} (1 - x)^{2} d x = \frac{2}{5}, E Y = \frac{2}{5} .

The mixed moment is

E [X Y] = \int_{0}^{1} \int_{0}^{1 - x} 24 x^{2} y^{2} d y d x = 8 \int_{0}^{1} x^{2} (1 - x)^{3} d x = \frac{2}{15} .

Thus the covariance is

Cov (X, Y) = \frac{2}{15} - \frac{4}{25} = - \frac{2}{75} .

The negative sign is consistent with the support constraint: a large value of one variable leaves a smaller available range for the other.

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