I was glancing at a probability and statistics review, and I saw some notation that I hadn't seen before. Naturally, I am curious as to how to properly read it and/or use it in the future. My question surrounds the use of the superscripted negative sign as follows:
"Discrete distributions can be specified by the probability mass function (pmf) $p_X(x) = P(X = x)$. The discontinuities in the cdf are at the points where the pmf is positive, and $p(x) = F(x) - F(x^-)$."
Please note, that I am wondering only about the notation - I am confident that I understand the functional and applied aspects of the above quoted passage.
Thanks for helping me learn a little more...
Usually, it's meant to describe the limit from the left (intuitively, pretend it was $x - \epsilon$ for some really small $\epsilon > 0$): $$ F(x^-) = \lim_{t \to x^-} F(t) $$
We don't usually see $F(x^+)$, since cdfs are right-continuous, so we could just call that $F(x)$ instead: $$ F(x^+) = \lim_{t \to x^+} F(t) = F(x) $$