Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 9 / b / v

Part (a) gives $\mathbb{P}(X\in \mathbb{Z})=0$. Therefore the ceiling function and floor function satisfy $\lceil X\rceil =\lfloor X\rfloor +1$ almost surely. Both are integrable, since $0\le \lfloor X\rfloor \le X$ and $\lceil X\rceil \le X+1$. Thus $\mathbb{E}[\lceil X\rceil ]=\zeta (t-1)+1$.

Now take expectations in $\lfloor X\rfloor \le X\le \lceil X\rceil$:Writing $s=t-1$, which ranges over every $s>1$, and rearranging yieldsThese are the zeta bounds from rounded Pareto variables: rounding below and above converts one elementary expected value into bounds on an infinite series.