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Let's consider a probability space $(\Omega, \mathcal{F}, P)$ corresponding to experiments on throwing a dice and defined in the following way: $\Omega = \{1, 2, 3, 4, 5, 6\}$, $\mathcal{F} = \{\Omega, \varnothing, \{1, 3, 5\}, \{2, 4, 6\}\}$. So, in this $\sigma$-algebra we only have 4 events: something happened, nothing happened, odd number rolled, even number rolled. Can anyone give me an example of random real variable defined for this probability space?

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A side remark: "dice" is the plural form, and the singular is "die". (The teacher in my first probability course at university said "we roll a die" with a heavy Indian accent a gazillion times, so I won't forget that!) – Hans Lundmark Jun 19 '11 at 20:37
A minor remark: the events $\Omega$ and $\varnothing$ do not describe the fact that something happened vs nothing happened. Rather they correspond to something certain happened vs something impossible happened (like, the result of the throw of the dice is an integer vs the result of the throw of the die is $7$). But something always happens. – Did Jun 27 '11 at 10:18
up vote 6 down vote accepted

Yes, $X(\omega)=0$ if $\omega \in \{1,3,5\}$, and $X(\omega)=1$ if $\omega \in \{2,4,6\}$.

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As a trivial example, you can take $X$ to be a constant random variable, for example $X(\omega)=1$ for all $\omega \in \Omega$. – Shai Covo Jun 19 '11 at 18:09
Thank you. Any other examples (of variables having other "structures")? – Leo Jun 19 '11 at 18:13
You can generalize as follows: $X(\omega)=a$ if $\omega \in \{1,3,5\}$, and $X(\omega)=b$ if $\omega \in \{2,4,6\}$. – Shai Covo Jun 19 '11 at 18:18
Note that the requirement is that for any $x \in \mathbb{R}$ fixed, $\{\omega: X(\omega) \leq x\} \in \mathcal{F}$. – Shai Covo Jun 19 '11 at 18:32
Or equivalently, since $\Omega$ is finite, for any $x \in \mathbb{R}$ fixed, $\{\omega:X(\omega) = x\} \in \mathcal{F}$. – Shai Covo Jun 19 '11 at 19:06

The random variables $X$ on this probability space are exactly the functions $X:\Omega\to\mathbb{R}$ such that there exists $a$ and $b$ with $X(\omega)=a$ if $\omega\in\{1,3,5\}$ and $X(\omega)=b$ otherwise. That is, every such $X$ is a random variable and every random variable $X$ is like that.

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Or in other words, $X$ has to be ${\mathcal F}$-measurable which means it can't depend on any other information than whether the result was even or odd. – Marek Jun 19 '11 at 18:22

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