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This theorem seems to hold great potential for an intuitive explanation, yet I'm still struggling for it.

Basically, it is saying that if $y = F(x)$, then y would have distribution $Unif[0,1]$. Intuitively, why?

Many thanks!

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What theorem are you speaking of? –  Ron Gordon Feb 3 '13 at 10:13
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If $F$ were invertible with inverse function $F^{-1}:(0,1)\to\mathbb R$, the event $[Y\leqslant y]=[F(X)\leqslant y]$ would coincide with $[X\leqslant z]$ with $z=F^{-1}(y)$, hence its probability would be $F(z)=y$. A random variable such that $\mathbb P(Y\leqslant y)=y$ for every $y$ in $(0,1)$ is uniformly distributed on $(0,1)$.

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