16. Under what conditions on $\xi$ are the random variables $\xi$ and $\sin \xi$ independent?
Solution. Let $y_{1}, \ldots, y_{n}$ be all the values ${ }^{3}$ that $\xi$ takes with positive probability. Then for any $i$ and $j, 1 \leqslant i0
\end{aligned}
$$

This means, $\sin y_{i}=\sin y_{j}$. Due to the arbitrariness of $i$ and $j$, we get $\sin y_{1}=\ldots=\sin y_{n}$, i.e.,

$$
\sin \xi=c \text { a.s., }
$$

where $c$ is a constant.

We will show that this is also a sufficient condition for the independence of the variables $\xi$ and $\sin \xi$. For any $x_{1}$ and $x_{2}$, consider the events $\left\{\xi=x_{1}\right\}$ and $\left\{\sin \xi=x_{2}\right\}$. We have $\mathrm{P}\left\{\sin \xi=x_{2}\right\}=1$, if $x_{2}=c$, and 0 otherwise. According to problem I. 3.6, the event $\left\{\sin \xi=x_{2}\right\}$ is independent of any event, in particular of $\left\{\xi=x_{1}\right\}$.