Question 196, Random variable $X(=1, 2, 3, \ldots), Y=(0, 1, 2)$, satisfy $P(X=k)=\frac{1}{2 k}$, and $Y \equiv$ $3(\bmod X)$, try to find $E(Y)$.
$$
\begin{array}{l}
E(Y)=0 \cdot P(X \equiv 0 \pmod{3}) + 1 \cdot P(X \equiv 1 \pmod{3}) + 2 \cdot P(X \equiv 2 \pmod{3}) \\
=1 \cdot \sum_{i=0}^{+\infty} \frac{1}{2^{3 i+1}} + 2 \cdot \sum_{i=0}^{+\infty} \frac{1}{2^{3 i+2}} \\
=\frac{1}{2} \cdot \frac{1}{1 - \frac{1}{8}} + 2 \cdot \frac{1}{2^2} \cdot \frac{1}{1 - \frac{1}{8}} = \frac{8}{7}
\end{array}
$$