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

For an integer $k\ge 1$, $\lfloor X\rfloor \ge k$ is the same event as $X\ge k$. Integration for the Pareto distribution givesThe second identity subtracts two successive tail probabilities, equivalently integrating over $[k,k+1)$. If $\mathbb{N}$ includes zero, the extra case is $\mathbb{P}(\lfloor X\rfloor \ge 0)=1$ and $\mathbb{P}(\lfloor X\rfloor =0)=0$, since $X\ge 1$ almost surely.