How can I understand the properties of a Cumulative Distribution Function (CDF)?

Question

Suppose that $F$ is the distribution function of a real-valued random variable $X$.

1. $F$ is increasing: if $x≤y$ then $F(x)≤F(y)$.
2. $F(x+)=F(x)$ for $x∈R$. Thus, $F$ is continuous from the right.
3. $F(x−)=P(X<x)$ for $x∈R$. Thus, $F$ has limits from the left.
4. $F(−∞)=0$.
5. $F(∞)=1$.

Let,

$Experiment$ = rolling a fair die.

So, $CDF = F(x) = \begin{cases} 0, & \text{if $0 \ge x$ or $6 < x$} \\ x \cdot \Bbb P(X=x), & \text{otherwise} \end{cases}$ (is it correct??)

then, how can I explain the above properties?

Answer 1

$X$ here is a discrete random variable, so the CDF of a discrete random variable is a summation. In this case: $\Pr(X \leq x) = \sum_{i=1}^6 \Pr(i)\mathbb{1}_{(i \leq x)}$.

You would integrate over the pdf of $x$, as it looks like you were trying to do, only if $X$ was a continuous random variable.

The + and - notation in the excerpt that you've quoted above is only to give us a framework for discussing atoms, or discontinuities in the density.