Probability Notation: What does $\{\omega\in \Omega : X(\omega) \in A\}$ mean?

Question

In the book Stats with Julia on p. 79 is reads ...

"The probability distribution of a random variable fully describes the probabilities of the events such as $\{\omega\in \Omega : X(\omega) \in A\}$ for all sensible $A \subset R$"

How would you say "$\{\omega\in \Omega : X(\omega) \in A\}$" in plain English?

Is it ....

for every possible outcome $(\omega)$ in the $(\in)$ event space $(\Omega)$ such that $(:)$ there is some specific outcome $(X(\omega))$ in the set $A$ where set $A$ contains real numbers

.. is that close??

Answer 1

[Edit] Yeah. The way to see, is to look to $X$ being a function (random variable). $$X: \Omega \longrightarrow \mathbb{R}.$$ The subset $A\subset \Bbb R$ is just to know what values can be and restrict the events $\omega\in \Omega$.

Answer 2

We have

• $\Omega$ = outcome space.

• $\omega$ = a particular outcome (that is, $\omega \in \Omega$).

• If $Z$ is an event then it is a subset of $\Omega$ (that is, $Z \subseteq \Omega)$. (*See footnote for an additional detail.)

Indeed the random variable $X$ is a function $X:\Omega \rightarrow \mathbb{R}$. Suppose $A$ is some given subset of real numbers. Then the following is a subset of $\Omega$: $$ \{\omega \in \Omega : X(\omega) \in A\} $$ We interpret this as:

\begin{align} \{\cdot\} \quad &= \quad \mbox{"The set of ..."}\\ \omega \in \Omega \quad &= \quad \mbox{"outcomes $\omega$ in the outcome space $\Omega$...}" \\ : \quad &= \quad \mbox{"such that..."}\\ X(\omega) \in A \quad &= \quad \mbox{"$X(\omega)$ is in the set $A$"} \end{align}

Put all together it reads:

The set of outcomes $\omega$ in the outcome space $\Omega$ such that $X(\omega)$ is in the set $A$.

Notice that $$ \{\omega \in \Omega : X(\omega) \in A\} \subseteq \Omega$$

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Example: \begin{align} \Omega &= \{blue, red, green, pink\}\\ X(blue) &= 2\\ X(red) &= 2.5\\ X(green) &=0\\ X(pink) &=-3\\ A &= \{2, -3, 8\}\\ B &= \{2.5, 0, -3\}\\ C &= \{x \in \mathbb{R} : x\leq 1\} = (-\infty, 1] \end{align} Then \begin{align} \{\omega \in \Omega : X(\omega) \in A\} &= \{blue, pink\}\\ \{\omega \in \Omega : X(\omega) \in B\} &= \{red, green, pink\}\\ \{\omega \in \Omega : X(\omega) \in C\} &= \: ??? \quad \quad [\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) \in A \cap B\} &= \: ???\quad \quad [\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) \notin A\} &= \: ???\quad \quad [\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) > 0\} &= \: ??? \quad \quad[\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) \leq 0\} &= \: ??? \quad \quad[\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) \leq 100\} &= \: ??? \quad \quad[\mbox{Exercise}]\\ \{\omega \in \Omega : X(\omega) \leq -78\} &= \: ??? \quad \quad[\mbox{Exercise}] \end{align}

How many possible events are there (for this example)?