= Solution
{c}
For an <integer> $k\geq1$, $\lfloor X\rfloor\geq k$ is the same event as $X\geq k$. Integration for the <Pareto distribution> gives
$$
\boxed{\mathbb P(\lfloor X\rfloor\geq k)=k^{1-t},\qquad
\mathbb P(\lfloor X\rfloor=k)=k^{1-t}-(k+1)^{1-t}.}
$$
The second identity subtracts two successive tail probabilities, equivalently integrating over $[k,k+1)$. If $\mathbb N$ includes zero, the extra case is $\mathbb P(\lfloor X\rfloor\geq0)=1$ and $\mathbb P(\lfloor X\rfloor=0)=0$, since $X\geq1$ almost surely.
Back to article page
