TITLE: What does "maximum value" of a set of random variables mean?
QUESTION [8 upvotes]: In our statistical mechanics lecture the professor said something along the lines of:

If we have some independent random variables $x_1,x_2,x_3,...,x_n$ having identical distributions:
Suppose $M_{n}=\text{max}(x_1,x_2,x_3,...)$, then we say that
  probability that $M_{n}<x$ is $\text{$\text{Pr}(M_{n}<x$)}$ (say).
In such a case
  $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

Now, first of all I don't understand what he meant by $M_{n}=\text{max}(x_1,x_2,x_3,...)$. What does maximum of a set of random variables even mean? Does it refer to the random variable which can take the highest value?
Secondly I don't understand the step $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

REPLY [3 votes]: (Looking just at the case with 2 random variables, for simplicity.)  
The maximum $\max(x_1,x_2)$ is itself a random variable, similar to e.g. the sum $x_1 + x_2$.  If $x_1$, $x_2$ are dice rolls, $\max(x_1,x_2)$ will be “the higher of the two values rolled”.
If you take any point in the sample space (informally: each time you run the experiment), you get actual values for $x_1$ and $x_2$.  So then you can take the sum of these values, $x_1 + x_2$; similarly, you can take their maximum, $\max(x_1,x_2)$.
So $x_1 + x_2$ and $\max(x_1,x_2)$ are themselves random variables — they don’t have a fixed value, but they get a value each time you run the experiment — formally, they’re mappings from the sample space to values.
Re your second question: the key logical property of the “maximum” function is that for any numbers $a_1, \ldots, a_n$,and $b$,
$$ \max(a_1,\ldots,a_n) \leq b \quad \Leftrightarrow \quad a_1 \leq b \land \ldots \land a_n \leq b. $$
I.e. the maximum is less than some ceiling exactly if all the individual values are less than that ceiling.  So this explains the first step in the probability calculation at the end.