 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/4/b/ii/solution

For an outcome

ω = (i, j)

, the two random variables are the coordinate maps

X (i, j) = i, Y (i, j) = j .

p_{X} (i) = p_{Y} (j) = 1/6

{1, \dots, 6}

p_{X, Y} (i, j) = \frac{1}{36} = p_{X} (i) p_{Y} (j) .

Off that support, the same factorization holds with zeros. Hence $X$ and $Y$ are independent random variables, by the joint probability mass function criterion for independence.

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