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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.

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  • $\begingroup$ $F(x)=\mathbb{P}(X\leq x)$. The set of outcomes $\omega$ such that $X(\omega)\leq -\infty$ is empty since there is no real number less than $-\infty$. $\endgroup$ – Nap D. Lover Nov 16 '19 at 23:19
  • $\begingroup$ In other words: for every real number $x\in \mathbb{R}$ we must have $-\infty <x<\infty$. $\endgroup$ – Nap D. Lover Nov 16 '19 at 23:30
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$F_X(-\infty)=P(X \leq -\infty)$ Sinec $X$ is areal number we cannot have $X \leq -\infty$. So $(X \leq -\infty)$ is the empty set.

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  • $\begingroup$ Why can't you have a number equal to -$\infty$? Or less? $\endgroup$ – ron jacobs Nov 16 '19 at 23:25
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    $\begingroup$ $-\infty$ is not a real number. @ronjacobs $\endgroup$ – Kavi Rama Murthy Nov 16 '19 at 23:26

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