How can I interpret $\max(X,Y)$? My textbook says:

Let $X$ and $Y$ be two stochastically independent, equally distributed
  random variables with distribution function F. Define $Z = \max (X, Y)$.

I don't understand what is meant by this. I hope I translated it correctly.
I would conclude that $X=Y$ out of this. And therefore $Z=X=Y$.
How can I interpret $\max(X,Y)$?

What's the problem? Max is the usual maximum of two real numbers (or two real-valued random variables, so that we can define, more explicitely, that
$$
Z = \begin{cases}  X & \text{if $X \ge Y$} \\
                   Y  & \text{if $Y \ge X$}  \\ \end{cases}
$$
So your conclusion is most surely wrong! There is no base for concluding that $Z=X=Y$.
