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ID: past-exam-of-the-options-i-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/n/1/c/solution

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Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / N / 1 / c / Solution by

Codex  0 2026-10-31

The five specified masses sum to one, so there are no other possible outcomes. Symmetry gives

E X = 0

, while

E Y = 1/4

. The two terms with

Y = 1

have opposite

X

values and equal masses, so

E (X Y) = \frac{1}{8} - \frac{1}{8} = 0, Cov (X, Y) = E (X Y) - E X E Y = 0.

Nevertheless

P (X = 0, Y = 1) = 0

, whereas

P (X = 0) P (Y = 1) = (1/2) (1/4) = 1/8

. Therefore $X$ and $Y$ are not independent. They are uncorrelated random variables, illustrating that zero covariance alone does not imply independence.

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