How to prove or intuitively understand that $\operatorname{P}(\max X_i > \varepsilon) = \operatorname{P}(\bigcup {X_i > \varepsilon})$ I am trying to at least get a feeling of how this equality works. I have a basic understanding of probability theory. However I could not wrap my head around this equality. Any explanation or a proof would help me.
If I understand this it might improve my understanding of the relationship between a random variable and seeing it as an event. Is that union of random variables a common thing to see? Don't we have union of events but joint dist. of random variables mostly?
Let $ X_1, \dots , X_n $ be $n$ random variables (can be independent).
$$\operatorname{P}\left( \max_{1\le i \le n} X_i > \varepsilon\right)= \operatorname{P}\left(\bigcup_i^n \{ X_i > \varepsilon\} \right)$$
I repeat what I read but it doesn't make sense to me even though maybe it is sensible. The probability of a maximum of a set random variable being bigger than epsilon is equal to the probability of the union of the set of random variables that are bigger than epsilon.
 A: If you know anything about measure theory, it may help some to expand out the part of the probability notation that people usually abbreviate. The LHS is
$$P\left ( \left \{ \omega \in \Omega : \max_{i=1,\dots,n} X_i(\omega)>\epsilon \right \} \right ).$$
The RHS is
$$P \left ( \bigcup_{i=1}^n \left \{ \omega \in \Omega : X_i(\omega) > \epsilon \right \} \right ).$$
Now if $\omega$ is such that $\max X_i(\omega)>\epsilon$, then there is some $i^*$ (depending on $\omega$) such that $X_{i^*}(\omega)>\epsilon$, in which case $\omega$ is in the $i^*$th set in the union. On the flip side, if $\omega$ is in the union, then there is some $i^*$ such that $X_{i^*}(\omega)>\epsilon$, and then the max is also bigger than $\epsilon$.
If you don't know anything about measure theory, suffice it to say that you should read "$\{ X_i>\epsilon \}$" as "the event that the random variable $X_i>\epsilon$", and "$\bigcup_{i=1}^n \{ X_i>\epsilon \}$" as "the event that at least one of the events "$X_i>\epsilon$" occurs". That is, $\{ X>\epsilon \}$ is not a set containing a random variable, it is an event, i.e. a set containing elements of the probability space.
A: Union means 'there exists'; intersection means 'for all'.
The definition of $x \in \bigcup_{\gamma \in \Gamma} A_{\gamma}$ is precisely $\exists\ \gamma \in \Gamma$ such that $x \in A_{\gamma}$.
The definition of $x \in \bigcap_{\gamma \in \Gamma} A_{\gamma}$ is $x \in A_{\gamma}\ \forall \gamma \in \Gamma$.
In your case, the max is greater than $\epsilon$ if and only if there exists one of them that is greater than $\epsilon$.
