Let $X$ be a continuous random variable with range $[x_l,\infty)$ and p.d.f.
$f_x(X)\propto x^{-a}$, for $x\in[x_l,\infty)$
for some values $x_l > 0$ and $a \in \mathbb{R}$.
Assume $x_l$ = 0.5. Let K = ceil(X) or floor(X), that is X rounded (Up or down) to the nearest integer.
i. State the range of K and derive its probability mass function p(k). Note that
Pk(K=k) = Px(k - 0.5 ≤ X < k + 0.5)
ii. Demonstrate that this equation for pk satisfies the requirements for a p. m. f.
iii. Without reference to the form derived in (i), please explain why (for small $a$)
p(k) = (2^(a-1)) * (a-1) * k^(-a)
Can someone please at least explain to me what exactly I need to do for all these steps? It's quite confusing as I have not found some proper explanations with regards to pmf anywhere. I would appreciate it even more if someone could help me solve them :)