24. Provide an example of two random variables $\xi$ and $\eta$, having the same distribution function ( $F_{\xi}=F_{\eta}$ ), but such that $\mathrm{P}\{\xi \neq \eta\}>0$.
Solution. Consider the sample space $\Omega=$ $=\left\{\omega_{1}, \omega_{2}\right\}, \mathrm{P}\left(\omega_{1}\right)=\mathrm{P}\left(\omega_{2}\right)=1 / 2$. Define the random variables pointwise: $\xi\left(\omega_{1}\right)=\eta\left(\omega_{2}\right)=1, \xi\left(\omega_{2}\right)=\eta\left(\omega_{1}\right)=-1$. Then $F_{\xi}(x)=F_{\eta}(x)$, but $|\xi-\eta| \equiv 2$.