It's been a while since I've dabbled in probability, so I'm just catching myself up on notation again. Currently, I'm working through Karlin and Taylor's A First Course in Stochastic Processes and am looking at question 2 in Elementary Problems.
The problem asks for a probability distribution of a discrete random variable $X$, then gives a solution stating $p(k) = \dots$.
My question is, does $p(k)$ mean $p(X = k)$ or $p(X \leq k)$? Based on my solution, I'm leaning towards the former, and assume this is due to the fact that $X$ is a discrete r.v. as opposed to a continuous (in which case a $p(X = k) = 0$). Is there a distinction between probability densities and distributions, and what notation $p(k)$ means in each case?