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I'd like to ask about middle of the discussion in this answer. Writing things out in reverse, $\mathrm{Prob}(F_X(X) \leq y) = \mathrm{Prob}(X \leq \mathrm{inf}\{x: F_X(x) \geq y\})$. Why is $\mathrm{inf}\{x: F_X(x) \geq F_X(X)\} = X$ (the left-hand sides of the 'less than or equal to' signs)? Wouldn't there still be an issue where there exists $x_0$ greater than the infimum but $F_X$ is the same for both?

(This answer composes the CDF with its inverse - the "These definitions imply..." bit. But I think that's easier than composing the CDF's inverse with the CDF.)

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  • $\begingroup$ Could you please make the question self-contained by quoting the parts of the earlier question that are relevant? $\endgroup$
    – Xi'an
    Mar 19 at 8:01

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