Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 8 / b / Solution

Complete the square in the joint probability density function:The density therefore factors as the standard normal distribution density for $Y$ times a normal distribution density in $x$ with mean $\rho y$ and variance $1-{\rho }^2$. It follows that $\mathbb{E}[Y]=0$, $\mathbb{E}[Y^2]=1$, and $\mathbb{E}[X\mid Y]=\rho Y$. Symmetry gives $\mathbb{E}[X]=0$. By the law of total expectation,Consequently the covariance of this bivariate normal distribution is $\mathrm{Cov}(X,Y)=\rho$. This calculation derives the result from the given density.