3

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??

StubbornAtom
  • 17,932
Axle Max
  • 221

2 Answers2

9

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

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)?

Michael
  • 26,378
  • *An additional property of events is that, not only are they subsets of $\Omega$, but they are in a pre-specified collection of such subsets that form a "sigma algebra." If $\Omega$ is a finite or countably infinite set then it is easiest to just assume the sigma algebra is the collection of all subsets of $\Omega$. Then all subsets of $\Omega$ are events, and so ${\omega \in \Omega : X(\omega) \in A}$ is indeed an event. – Michael Jun 17 '19 at 04:23
  • But what is A in your case? You never defined it. – nonuser Jun 17 '19 at 05:14
  • 1
    The asker Axle never defined the set $A$, presumably it is a given subset of real numbers. In my specific example I gave a specific set $A$. I have modified my answer above to emphasize that $A\subseteq \mathbb{R}$ also in the first part. – Michael Jun 17 '19 at 05:15
  • So we can use $X^{-1}(A)$ notation? – nonuser Jun 17 '19 at 05:19
  • You could, indeed by definition $$X^{-1}(A)={\omega \in \Omega : X(\omega) \in A}$$ – Michael Jun 17 '19 at 05:21
2

[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 $\omega\in \Omega$.

Gustavo Mezzovilla
  • 823
  • Thanks. What does the colon mean please? – Axle Max Jun 17 '19 at 02:15
  • 1
    I don’t agree with the last line you’ve mentioned. “if you want all the events of omega that only have greater or equal to 90% chance occurring can be described by ________”. To describe it correctly you need to define a “sigma-algebra” and a Probability Measure. – Vishaal Sudarsan Jun 17 '19 at 03:46
  • 1
    It is a notation to describe a set. You can read it like such as somenthing. Normally, a set is given by all the elements that lives on a bigger set SUCH satisfy some property.

    I recomend you to see some inital set theory to a better explanation. You will see stuffs like ${x\in X : P(x)}$, i.e., all the elements of $X$ wich holds the property $P$. Like the even numbers is given by ${n \in \mathbb{Z} : 2\mid n}$. The property here is $P(n) = 2\mid n$, "2 divides $n$".

     – Gustavo Mezzovilla Jun 17 '19 at 03:49
  • @VishaalSudarsan Sure, sure. But i think that the discussion are more about the notation, so i was trying to give an exemple to ilustrate. That preciosity and rigour's do not are needed to the point here (im my opinion). But sure, we would need $(\Omega,\mathcal{F},X)$ where $\mathcal{F}$ is an $\sigma$-algebra and $X$ satisfy Kolmogorov condictions. – Gustavo Mezzovilla Jun 17 '19 at 04:09
  • 3
    You're still not correct $X(\omega)$ does NOT represent the probability of event $\omega$ happening. $X(\omega)$ is just a function to the real line. It can take any real number. – Vishaal Sudarsan Jun 17 '19 at 04:16
  • Please correct or remove the false statement that $X(\omega)$ represents the probability that $\omega$ occurs. Take any non-unit interval $[a, b]$ say with $1<a<b$ and define $X(\omega)=\omega$. Then the values of $X(\omega)$ cannot represent probabilities, while the events $\omega\in (a, x]$ have chance $\omega/(b-a)$. Further, normally distributed random variable can take on negative values making your statement even more nonsensical. – Nap D. Lover Aug 07 '21 at 10:42