Exactly $\omega$ $\in$ $\Omega$

measure-theoryprobabilityprobability theoryunit-of-measure

Example:

We toss a coin 3 times:

$\Omega$ = {$\omega_1 \omega_2 \omega_3$} = {HHH,HHT,HTH,TTT,THT,TTH,TTT}

2 times:

$\Omega$ = {$\omega_1 \omega_2$} = {HH,HT,TH,TT}

1 time:

$\Omega$ = {$\omega_1$} = {H,T}

What the heck does $\omega_1$ mean? I've read elsewhere that it's a singleton but really I don't know.

Does it mean:

{$\omega_1 \omega_2$} = {$\omega_1 *\omega_2$} = {H,T} * {H,T}

If we say that X is a stockprice that has on time $X_1$ three outcomes namely {Up(=U), Stays the same price(=EQ), Down(=D)} is

$\Omega$ = {$\omega_1$} = {U,EQ,D}

On time $X_2$

$\Omega$ = {$\omega_1 \omega_2$} = {U,EQ,D}*{U,EQ,D} = {UU,UEQ,UD,EQU,EQEQ,EQD,DU,DEQ,DDD}

I have looked at quite some answers like:

What is $\omega$ in probability theory?

Rigorous Meaning of "Drawing a Sample" $\omega$ from a Probability Space $(\Omega, \mathcal{A}, \mathbb{P})$

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

But still I don't understand. Can someone please give an example with real numbers like a stock price or the temperature with if not's too much work a graph?

Best Answer

What the heck does $\omega_1$ mean?

Here $\omega_k$ is being used to represent "the result of the $k$th coin toss".

As JMoravitz commented, your source is taking the notation $\{\omega_1\omega_2\omega_3\}$ to represent the outcome set generated by the first three coin toss results. That is, as an abbreviation for $\{\langle\omega_1,\omega_2,\omega_3\rangle\in\{H,T\}^3\}$

So, yes, that does mean $\{\omega_1\omega_2\}$ is the cartesian product $\{\omega_1\}\times\{\omega_2\}$

So when instead using your stockmarket exsample, with the options $U,E,D$. (Advise: don't use two letters when one will do, especially when concatenating them into strings.)

$$\begin{align}\{\omega_1\omega_2\}&=\{U,E,D\}^2\\&=\{UU,UE,UD,EU,EE,ED,DU,DE,DD\}\end{align}$$

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