Calculating the probability mass function 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 :)
 A: $X$ is a continuous random variable 
$K$ is a discrete random variable formed by rounding $X$ to the nearest integer.  So for example when $0.5 \le X \lt 1.5$ then $K=1$.  You can find the probability $K$ takes a particular value by integrating the density function for $X$ between suitable limits.  This will give you the proability mass function.
You can show it is a probability mass function if each probability is non-negative and their sum is $1$.
Part (iii) is slightly strange with "Without reference to the form derived in (i)".  You need to find an alternative approach: induction would give you the $k^{-a}$ term.
A: For part (iii).
From  this post, you have
$$
f_X(x) =2^{a-1}(a-1)x^{-a}, \ x\ge 1/2.
$$
Then
$$\eqalign{
p(k)&= \int_{k-{1\over2}}^{k+{1\over2}} 2^{a-1}(a-1)x^{-a}\,dx\cr 
&\approx
\Bigl(  (k+{1\over2}) - (k-{1\over2})   \Bigr) 2^{a-1}(a-1)k^{-a}\cr
&
= 2^{a-1}(a-1)k^{-a}.}
$$
I'm not sure if this is what you're expected to do, as the integral above can be computed exactly (and this is what you do for part (i)).
Perhaps you're expected to think of  the pmf  as a "bar chart". The bar chart can be approximated by the graph of the density function. Each bar has width 1, is centered at $k$, and the height is $f_X(k)= 2^{a-1}(a-1)k^{-a}$.
(I'm not sure what "for small $a$" means here, $a$ must be greater than 1 in order for $f$ to actually define a density.)
