Discrete Random Variable

Definition We say a random variable $\mathbb{X}$ is discrete if the range of $\mathbb{X}$ is at most countable, and we define probability mass function (p.m.f) of $\mathbb{X}$ to be $p(x) = \mathbb{P}\{\mathbb{X}=x\}$, denoted $\mathbb{X} \sim p(x)$

This is very straight forward, because $\mathbb{P}\{\mathbb{X}=x\}$ is, well, $\mathbb{P}\{\mathbb{X}=x\}$. There really is no need for other fancy method.

Remark

If $\mathbb{X}$ is discrete r.v with p.m.f $p(x)$ and c.d.f $F(x)$

(a) The sample space $S$ need not be at most countable, only the range of $\mathbb{X}$ need to be at most countable.

(b) $p(x) > 0$ for at most countable point, because the range of $\mathbb{X}$ is at most countable.

(c) $F(x) = \sum_{t\leq x, p(t)>0}p(t)$

Here, we cannot simply sum $p(t)$ over all $t\leq x$, because the number of $t\leq x$ might be uncountable.