Let $X \geq 1$ be a integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous position. So the question is, if we stop this process after the first time $\tau$ for which $Y_{\tau}=\sum_{i=1}^{\tau}X_i > K$, that is, after the first time it crosses $K$, then what is $E[Y_{\tau}-K]$?.   Can we get a bound of $O(\mu)$?