Explain the following statements about random variables. For $E \subseteq X(\Omega)$, respectively for $x \in X(\Omega)$ we can write:


*

*$\{X \in E\} := \{\omega \in \Omega : X(\omega) \in E\} = X^{-1}(E)$

*$\{X = E\} := \{\omega \in \Omega : X(\omega) = x\} = X^{-1}(\{x\})$

*$\{X \leq x\} := \{\omega \in \Omega : X(\omega) \leq x\} = X^{-1}(]-\infty,x])$
I'm not sure what some of the symbols actually mean. Since $X(\Omega)$ is the range of the random variable $X$, is $E$ then a subset from said range? If that is the case, then is $x$ a specific value from the range of $X$?
Additionally, if the statements can be explained in "plain" language that would be very helpful.
 A: A random variable is just a special type of function, so forget for a moment that it's a random variable and just pretend it's just another function, say $f$. Given a function $f : Y \to Z$ and $B \subseteq Z$, you are probably familiar with this:
\begin{align*}
f^{-1}(B) = \{y \in Y : f(y) \in B\}\\
\end{align*}
Thus, for the first statement, it is simply all the events $\omega \in \Omega$ in the sample space such that $X(\omega) \in E$. For the second and third, they're simply a special case of the first, with $E = \{x\}$ and $E = (-\infty,x]$ respectiely.

If you would like a plain language for the first statement (which is also the same reasoning for the second and third), a random variable $X$ "encodes" events in $\Omega$ in the form of real numbers. For instance, in the case of roulette, I can have $X$ defined as follows:
\begin{align*}
X = 
\begin{cases}
1, &\text{if $\omega$ is a red number} \\
-1, &\text{if $\omega$ is a black number} \\
0, &\text{if $\omega = 0$} \\
\end{cases}
\end{align*}
Thus, here I am encoding the roulette numbers by their colour. $X^{-1}(E)$, where $E \subseteq \mathbb{R}$, would thus gives us the set of events in which, when encoded, lies inside the set $E$. For instance, for $E = \{\pm 1\}$, $X^{-1}(E)$ asks for the events which, when encoded, is non-zero, which would be all numbers on the roulette except $0$.
