# How to prove the probability in copula and what does it means

I am learning copula. However, it seems quite hard to understand its theory. I found a useful, but with some unclear statements, post. This post can be accessed from here. the link.

I just cannot understand the proof of this formula.

$x_1, x_2$ are two random variables.

$u_1 = F(x_1)$, $u_2 = F(x_2)$

Then,

$F(u) = P(U \leq u)$ = $P(F(x) \leq (u))$ ## this is ok for me.

$= P( F^{-1} (F(x)) \leq F^{-1} (u))$

My question is: From where $F^{-1}$ come from?

The second part of my question is,

"The above statements can be summarized as saying that the values of the CDF of any marginal distribution are uniformly distributed on the interval [0,1] ie. if you make a random draw from any distribution, you have the same probability of drawing the largest value (U=1) of that distribution as the smallest possible value (U=0) or the median value (U=.5)."

What does this mean? Could someone explain it in a simple way, please?

For the first part of your question, and if you are referring simply to the usage of $F^{-1}$ in that step, it comes from the assumption that $X$ is a random variable with a continuous CDF, i.e. the function $\mathbb{P}[X\leq x] = F(x)$ is continuous, thus the usage of $F^{-1}$ is not a problem.
If you are instead wondering why it appears in the last part of the equality, the events $\{F(X)\leq u\}$ and $\{F^{-1}(F(X))\leq F^{-1}(u)\}$ are equivalent (and therefore their probabilities are equal) because $F$ is continuous and, as a df, also an increasing function, thus its inverse, $F^{-1}$, is also increasing.
For the second part, the text you quote is just a very messy and quite misleading way of describing the fact that $U:=F(X)\sim\mathcal{U}(0,1)$. A better way to phrase it would be that if we draw a sample $x_1,\ldots,x_n$ from any distribution with df $F$, and calculate $F(x_i) = u_i$, $i=1,\ldots,n$, then these $u_i$ are samples from a uniform distribution. By itself, this might not say much to you, but it's very useful since we can use it the other way around as well: just as we can transform the outcomes of any distribution with df $F$ into a sample of $\mathcal{U}(0,1)$, we can also transform any outcomes of $\mathcal{U}(0,1)$ to a sample of a distribution with df $F$ by use of the inverse of $F$ through the relationship $F(x_i) = u_i\Leftrightarrow F^{-1}(F(x_i)) = F^{-1}(u_i) \Leftrightarrow x_i = F^{-1}(u_i)$. This procedure is known as the quantile (or inverse probability integral) transform.