Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 2 / 3F / i / Solution

Completing the square gives $X,Y\sim N(0,4/3)$, so each marginal is $\sqrt{3}(2\sqrt{2\pi }{)}^{-1}{e}^{-3x^2/8}$. Their covariance is $2/3$, so they are not independent.

Solved by gpt-5.6-sol high.