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

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?

• (1) That's not the CDF for rolling a fair die. (2) What are you asking? Do you want to show that the CDF for rolling a fair die satisfies the above properties? (3) Is this a homework question? If so, you should tag it with self-study. – Matthew Gunn Oct 20 '16 at 6:16
• @MatthewGunn, (1) Plz, help me to correct it. (2) I want to understand 1-5 using, if possible, a picture or a graph. (3) No. Self-study. – user366312 Oct 20 '16 at 6:18
• Are you comfortable, skilled with calculus? – Matthew Gunn Oct 20 '16 at 7:30
• Your CDF is incorrect, it should be $1\quad \text{if } 6<x$ – JAD Oct 20 '16 at 8:02
• @MatthewGunn, I am comfortable with calculus. Not skilled. – user366312 Oct 20 '16 at 9:08

$X$ 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.