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I have a cumulative distribution function $F(x)= \mathbb P(X < x)$.

Now, for some further purposes, I need to find its inverse. The case is also that I am dealing with a KS test, where I have the implication: $\sup |F_n(X) - F(X)|$. So, what does it mean if I define the inverse of $F$ as

$$ F^{-1}(y)= \min \{F(x)>y\} \>? $$

What does this represent visually? And why is it $F(x)>y$ that we are looking at? (I can see how it comes about algebraically, but I cannot visualise it).

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The cumulative distribution function is usually defined as $F(x)=\Pr(X \le x)$, i.e. is right continuous. – Henry Oct 23 '11 at 18:54
Related:… – cardinal Oct 23 '11 at 22:19
up vote 3 down vote accepted

For a visualisation try this with the CDF and its reflection

enter image description here

and note the vertical green line segments which strictly are not parts of the functions but appear as horizontal line segments in the reflections.

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