What is an expression $ x ^ - $ and which of these properties are applicable to a continuous random variable as well?

Question

Properties:

-Apply to distribution functions of discrete random variables-

• $\operatorname {P}(X
• $\operatorname {P}(X>a)=1-F(a)$
• $\operatorname {P}(X\geq a)=1-F(a^{-})$
• $\operatorname {P}(a
• $\operatorname {P}(a\leq X
• $\operatorname {P}(a\leq X\leq b)=F(b)-F(a^{-})$

I would like to know:

• What does the expression "$x^-$" mean?
• And which of the properties listed above apply to continuous random variables as well?

Thank you very much.

Answer 1

$F(a^-)$ is the limit from the left (or "from below").

$$F(a^-) ~=~ \lim\limits_{x\to a^-} F(x) ~=~ \lim\limits_{x\nearrow a} F(x) ~=~ \lim_{a> x\to a} F(x)$$

Similarly $F(x^+)$ is the limit from the right (or "from above").

$$F(a^+) ~=~ \lim\limits_{x\to a^+} F(x) ~=~ \lim\limits_{x\searrow a} F(x) ~=~ \lim_{a

In particular, a CDF must be right-continuous, that is $F(a)~=~ \mathsf P(X\leq a)~=~F(a^+)$, and $F(a^-)~=~\mathsf P(X