(a) Let $X$ and $Y$ be independent random variables taking values $\pm 1$, each with probability $\frac{1}{2}$, and let $Z=X Y$. Show that $X, Y$ and $Z$ are pairwise independent. Are they independent?

(b) Let $X$ and $Y$ be discrete random variables with mean 0 , variance 1 , covariance $\rho$. Show that $\mathbb{E} \max \left\{X^{2}, Y^{2}\right\} \leqslant 1+\sqrt{1-\rho^{2}}$.

(c) Let $X_{1}, X_{2}, X_{3}$ be discrete random variables. Writing $a_{i j}=\mathbb{P}\left(X_{i}>X_{j}\right)$, show that $\min \left\{a_{12}, a_{23}, a_{31}\right\} \leqslant \frac{2}{3}$.