 Solution

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

For

0 \leq x \leq 1

y

-range:

f_{X} (x) = \int_{0}^{1 - x} 24 x y d y = 12 x (1 - x)^{2} .

Outside this interval it is zero. Therefore the marginal density is

f_{X} (x) = {12 x (1 - x)^{2}, 0, 0 \leq x \leq 1, otherwise .

12 (1/2 - 2/3 + 1/4) = 1

, also checking the original joint normalization. By symmetry

Y

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