Properties:
-Apply to distribution functions of discrete random variables-
- $\operatorname {P}(X<b)=F(b^{-})$
- $\operatorname {P}(X>a)=1-F(a)$
- $\operatorname {P}(X\geq a)=1-F(a^{-})$
- $\operatorname {P}(a<X<b)=F(b^{-})-F(a)$
- $\operatorname {P}(a\leq X<b)=F(b^{-})-F(a^{-})$
- $\operatorname {P}(a\leq X\leq b)=F(b)-F(a^{-})$
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.
$F(a^-)$ 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").
$$F(a^+) ~=~ \lim\limits_{x\to a^+} F(x) ~=~ \lim\limits_{x\searrow a} F(x) ~=~ \lim_{a<x\to a} F(x)$$
In particular, a CDF must be right-continuous, that is $F(a)~=~ \mathsf P(X\leq a)~=~F(a^+)$, and $F(a^-)~=~\mathsf P(X<a)$.