Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 6 / b / iv

For a nonnegative integer-valued random variable $W$, the tail-sum formula states $\mathbb{E}[W]={\sum }_{k=1}^{\infty }\mathbb{P}(W\ge k)$. It follows by writing $W={\sum }_{k\ge 1}{1}_{\{W\ge k\}}$ and using nonnegativity to interchange sum and expectation.

With $W=\lfloor X\rfloor$, this givesThe Riemann zeta function is finite here because $t-1>1$.