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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?

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    $\begingroup$ (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. $\endgroup$ – Matthew Gunn Oct 20 '16 at 6:16
  • $\begingroup$ @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. $\endgroup$ – user366312 Oct 20 '16 at 6:18
  • $\begingroup$ Are you comfortable, skilled with calculus? $\endgroup$ – Matthew Gunn Oct 20 '16 at 7:30
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    $\begingroup$ Your CDF is incorrect, it should be $1\quad \text{if } 6<x$ $\endgroup$ – JAD Oct 20 '16 at 8:02
  • $\begingroup$ @MatthewGunn, I am comfortable with calculus. Not skilled. $\endgroup$ – user366312 Oct 20 '16 at 9:08
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$X$ 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.

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