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 $\max(X,Y)$?
 A: One concrete example:
Suppose each of the four cells below has probability $1/4$:
$$
\begin{array}{|c|c|}
\hline X=0,\ Y=0 & X=1,\ Y=0 \\  \hline X=0,\ Y=1 & X=1,\ Y=1 \\  \hline
\end{array}
$$
Then here is how $\max\{X,Y\}$ is distributed:
$$
\begin{array}{|c|c|}
\hline \max=0 & \max=1 \\  \hline \max=1 & \max=1 \\  \hline
\end{array}
$$
Each cell still has probability $1/4$, so $\Pr(\max\{X,Y\}=1) = 3/4$.
A: 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 $Z=X=Y$.
A: A random variable is a function from a sample space to $\mathbb{R}$. Therefore $Z=\max(X,Y)$ means $Z(p):=\max(X(p), Y(p))$ for each $p$ in the sample space.
Also, from that you can't conclude $X=Y=Z$.
A: Think of it this way: to sample from $Z$, take one sample point from $X$ ($x_i$) and one from $Y$ ($y_i$). The point you sampled from $Z$ is $\max(x_i,y_i)$.
If by $X = Y$ you mean that $X$ and $Y$ are the same distribution AND they're perfectly correlated, then indeed $X = Y = Z$. Otherwise, no. Take for example $X$ and $Y$ to be independent, uniform distributions on $[0, 1]$. Then $Z = \max(X, Y)$ will follow a triangular distribution: $P(Z = z_i) = 2z_i$.
