All cumulative distribution function follows a U[0,1] In my stats lecture, my professor introduced this theorem, however I don't quite understand what the theorem means and how he got from step 3 to 4. Also what does the prime symbol mean? Could someone paraphrase what the theorem means or point me to some online resource where I could learn more about this theorem? Thanks
Theorem: If $X$ is a continuous random variable with cumulative distribution function (CDF) $F$, then $Y = F(X) \sim U[0,1]$
Proof:
Let CDF of $Y$ = $F_y$
$$F_y=P(Y \leq y)$$
$$=P(F(X) \leq y)$$
$$=P(X \leq F^{-1}(y))$$
$$=F(X^{'})$$
$$=F(F^{-1}(y))=y$$
$$Y \sim U[0,1]$$
 A: The proof you cite glosses over a technical point which arises when the distribution function $\ F\ $ is not strictly increasing.  In that case, it is not injective (one-to-one), so it's not altogether clear what the definition of $\ F^{-1}\ $ should be.  However, the last step of the proof only requires that $\ F\left(F^{-1}(y)\right)=y\ $—that is, that $\ F^{-1}\ $ be a left inverse of $\ F\ $. The other essential property needed (in the second step of the proof) is that
$$
F(x)\le y \iff x\le F^{-1}(y)\ .
$$
The function $\ F^{-1}\ $ defined by
$$
F^{-1}(y)=\inf\left\{x\,| y\le F(x)\right\}
$$
has both of these properties, so if you take that as its definition then the proof holds up.
A: This theorem is named "integral transformation theorem "
As you can see after few passages you get
$$F_Y(y)=F_X[F_X^{-1}(y)]$$
now it is evident that applying both $F$ and $F^{-1}$ to any $y$ they cancel each other, say $e^(log y)$ thus yoy get
$$F_Y(y)=y$$
But $F=y$ is the CDF of the uniform distribution over $[0;1]$
Derivate it and get
$$f_Y(y)=\mathbb{1}_{[0;1]}(y)$$
This theorem is very useful for many purposes, for example using it backwards you can generate a random sample from any continuous distribution starting from a random sample extracted by a $[0;1]$ uniform distribution.
Example
Suppose you have to generate a random sample by an Exponential $Exp(1)$ distribution, say
$$F_X(x)=1-e^{-x}$$
By integral transformation theorem you know that
$$y=1-e^{-x}\sim U[0;1]$$
and thus being
$$x=-log(1-y)$$
you can have your random number $x$ using the uniform random number $y$ that is automatically generated by any calculator, also in EXCEL and often also by a poket calculator
