In the proof of the probability integral transform at Wikipedia, they take any real-valued random variable $X$ and use the inverse of its distribution function $F_X$.
However, a distribution function is monotonically increasing, but not necessarily strictly increasing. So, the inverse of $F_x$ might not even exist.
Is this an error from Wikipedia or do I miss something?
Best Answer
Because $F_X$ need not be strictly increasing, one must use a sort of generalized inverse. Preferences vary with authors, but here is one example (the "right-continuous inverse"): $$ F^{-1,+}_X(t):=\inf\{x: F_X(x)>t\}, \quad t\in[0,1], $$ wherein $\inf\emptyset$ is understood to equal $1$.
For example, if $X$ has cdf $$ F_X(x)=\cases{0,&$x<1$;\cr {1\over 3},&$1\le x<2$;\cr 1,&$x\ge 2$,\cr} $$ then $$ F^{-1,+}_X(t) =\cases{1,&$0\le t<{1\over 3}$;\cr 2,&${1\over 3}\le t\le 1$.\cr} $$ In this case, if $U$ has the uniform distribution on $[0,1]$, then $F^{-1,+}(U)$ takes the value 1 with probability $1/3$ and the value 2 with probability $2/3$, so $F^{-1,+}(U)$ has the same distribution as $X$.
The "left-continuous inverse" $F^{-1,-}_X(t):=\sup\{x: F_X(x)\le t\}$ would work as well.