Suppose that $F$ is the distribution function of a real-valued random variable $X$.
- $F$ is increasing: if $x≤y$ then $F(x)≤F(y)$.
- $F(x+)=F(x)$ for $x∈R$. Thus, $F$ is continuous from the right.
- $F(x−)=P(X<x)$ for $x∈R$. Thus, $F$ has limits from the left.
- $F(−∞)=0$.
- $F(∞)=1$.
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?