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$F$ is cdf of r.v. $X$. Define $\tilde{X} := F(X^−)$, where $F(x_0^-) := \lim{x \to x_0^-} F(x)$. This is what I sometimes see from statistic books especially about nonparametric statistics.

I haven't been feeling comfortable with $F(X^−)$ yet. I was wondering if there is some equivalent way to represent $F(X^−)$?

I though it was equivalent to $P_{X'}(X' < X)$, where $X'$ and $X$ are iid with cdf $F$, and $P_{X'}$ means taking probability wrt $X'$. But I later realized they are not equivalent.

In principle, when is $F(X^−)$ used and when is $P_{X'}(X' < X)$ used instead?

Thanks and regards! Thank and regards!

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