problem 43

(a): Let $X \equiv a$ (mod $b$ ) and $Y \equiv X + 1$ (mod $b$ ). Then, $lim b \to \infty P (X < Y) = 1$ .
For finite random variables $X$ and $Y$ , the case of $P (X < Y) \geq 1$ is not possible, since then $Y$ can never achieve the smallest value of $X$ , contradicting the assumption that $X$ and $Y$ have the same distribution.
(b): If $X$ and $Y$ are independent random variables with the same distribution, then $P (X < Y) \leq \frac{1}{2}$