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?
Suggest an example of random variable
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probability-theory
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3A 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!) – 2011-06-19
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2A 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*. – 2011-06-27
2 Answers
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Yes, $X(\omega)=0$ if $\omega \in \{1,3,5\}$, and $X(\omega)=1$ if $\omega \in \{2,4,6\}$.
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1As a trivial example, you can take $X$ to be a constant random variable, for example $X(\omega)=1$ for all $\omega \in \Omega$. – 2011-06-19
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0Thank you. Any other examples (of variables having other "structures")? – 2011-06-19
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0You can generalize as follows: $X(\omega)=a$ if $\omega \in \{1,3,5\}$, and $X(\omega)=b$ if $\omega \in \{2,4,6\}$. – 2011-06-19
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0Note that the requirement is that for any $x \in \mathbb{R}$ fixed, $\{\omega: X(\omega) \leq x\} \in \mathcal{F}$. – 2011-06-19
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0Or equivalently, since $\Omega$ is finite, for any $x \in \mathbb{R}$ fixed, $\{\omega:X(\omega) = x\} \in \mathcal{F}$. – 2011-06-19
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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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3Or 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. – 2011-06-19