# Cumulative distribution function property

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.

• $F(x)=\mathbb{P}(X\leq x)$. The set of outcomes $\omega$ such that $X(\omega)\leq -\infty$ is empty since there is no real number less than $-\infty$. – Nap D. Lover Nov 16 '19 at 23:19
• In other words: for every real number $x\in \mathbb{R}$ we must have $-\infty <x<\infty$. – Nap D. Lover Nov 16 '19 at 23:30

$$F_X(-\infty)=P(X \leq -\infty)$$ Sinec $$X$$ is areal number we cannot have $$X \leq -\infty$$. So $$(X \leq -\infty)$$ is the empty set.
• Why can't you have a number equal to -$\infty$? Or less? – ron jacobs Nov 16 '19 at 23:25
• $-\infty$ is not a real number. @ronjacobs – Kavi Rama Murthy Nov 16 '19 at 23:26