There are two propositions I'm struggling with interpreting what they really mean. Perhaps I'm expecting more even though they appear simplistic:
$X$ is defined as a random variable, with density $f$ and cdf $F$. With the appropriate conditions to make them "nice".
First is my understanding of Proposition C. Z is usually reserved for a standard normal random variable and I took it as such. So in my mind what proposition C is saying is "Let our standard normal random variable be a function of an arbitrary CDF. Then our standard normal random variable has a uniform distribution on $[0,1]$." But here is where it get's cloudy for me…..we have this standard normal random variable, this standard normal random variable is going to have a distribution, since it has been declared as a "standard normal random variable" I would assume that its distribution would be "normally distributed". The values for this standard normal random variable, $Z$, happen to come from the CDF of some other random variable, $X$. I see the proof and the steps of the proof are very simple and sound, but the concept is not clicking with me.
With regards to Proposition D I understand what that is saying. In practice we would start with a CDF, $F(x)$ and rearrange things in a way to apply a uniform random generator.
Help with interpreting Proposition C?
Best Answer
In proposition C and D, what is missing are the assumptions.
Proposition C should be stated: If $X$ is a random variable with continuous CDF $F_X$ then random variable $Z:= F_X(X)$ is uniformly distributed on $(0,1)$ (it's the same as uniformy distributed on $[0,1]$)
The proof that is in the book, assumes that $F$ has inverse $F^{-1}$ (just to have simplier proof, but it isn't necessary for proposition C to hold). Moreover, in the proof, it should be said that those equations are true for $z \in [0,1]$, otherwise we have for $z<0$: $\mathbb P(Z \le z) = \mathbb P(F_X(X) \le z) = 0$ (since $F_X$ is a function with values in $[0,1]$) and for $z>1$ we have $\mathbb P(Z \le z) = \mathbb P(F_X(X) \le z)=1$, since as we said before, function $F_X$ always takes values less or equal to $1$.
Proposition $D$ is okay, but we need to know that $F$ is right continuous function, non-decreasing with values in $[0,1]$. Proposition assumes that $F^{-1}$ exists, which again isn't neccesary, but then you need the notion of generalised inverse to write the proposition $D$.