Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 7 / b / iii / Solution

The marginal distributions have means $\mathbb{E}[X]=p$ and $\mathbb{E}[Y]=p$. In the joint distribution, $XY$ is nonzero only at $(1,1)$: when $X=1$, the only possible values of $Y$ are $0$ and $1$. HenceThese quantities are finite; for example, $0⩽XY⩽1$. The covariance changes sign when $p=e^{-p}$.