In my notes I have that one of the properties of the cumulative distribution function is,
(1) $\lim_{x \to -\infty}F_X=0$ , $ \lim_{x \to \infty}=1.$
Then it keeps on saying that intuitively this is true because $F_X(-\infty)=\mathbb{P}(\emptyset)=0$ and $F_X(\infty)=\mathbb{P}(\mathbb{R})=1$.
I'm not interested in a formal proof of (1). I understand that if the sample space is the real numbers then $F_X(\infty)=\mathbb{P}(\mathbb{R})=1$,the probability of the sample space is always one.
But I don't quite understand why does it says that that the $cdf$. at $-\infty$ equals the probability of the emptyset. $-\infty$ means that the numbers don't stop decreasing , but you will always have real numbers and not an emptyset.
Best Answer
$F_X(-\infty)=P(X \leq -\infty)$ Sinec $X$ is areal number we cannot have $X \leq -\infty$. So $(X \leq -\infty)$ is the empty set.