What Is the Domain of a Die?

[Rasalhague]

What is the "domain of a die"?

Consider a fair dice [sic] with six faces. The random variable X is the dice. The alphabet $\cal{X}$ is the set {1,2,3,4,5,6}.

- Haim Permuter: Mathematical methods in communication: http://www.ee.bgu.ac.il/~it09/lec1/lec1.pdf (PDF notes).

What is the domain of X, and what is its rule?

Is the domain the set of ideas {the concept of the die landing on 1,...,the concept of the die landing on 6}, and is the rule X(the concept of the die landing on x) = x, x $\in$ {1,2,3,4,5,6}? Or is X just the identity function on {1,2,3,4,5,6}? Or could the domain be anything, so long as it and its associated probability measure are such that X is measurable?

 

[pwsnafu]

A normal die is probably not the best example because the numbers "1", "2", and so on are written on most dice. Some dice use pips, but I don't think physics forums has a LaTeX package which let's us draw pips (I think it was called epsdice ro something).

So I going to use Fate dice for this example. If you don't know, a Fate die, written dF for short, is a six sided die with a plus symbol on two faces, a minus sign on two faces, and a mathematical dot on the remaining two faces.

Now the state space is these symbols. Because each is repeated, we should technically distinguish them. So $\Omega = \{ {+}_1, {+}_2, {\cdot}_1, {\cdot}_2,{-}_1, {-}_2 \}$. Now the random variable $X : \Omega \rightarrow \Rset$ is

• $X(-_1) = -1,$
• $X(-_2) = -1,$
• $X(\cdot_1) = 0,$
• $X(\cdot_2) = 0,$
• $X(+_1) = 1,$
• $X(+_2) = 1.$

What a random variable does is that it takes values (which might not be numbers) and represents each element as a number. This allows us to say calculate the standard deviation of a Fate die, even though there are no numbers written on it.

 

[Rasalhague]

Thanks for the answer, pwsnafu. (So, in the case of pips, if we identify the pips with the numbers they denote, and identify the die with this random variable, then I guess, yes, it could well be the identity function.)

First, just to avoid confusion, I gather random variable has two senses: (1) a measurable function; (2) a real-valued measurable function; and I think you're using it in sense 2.

Second, is there some special, distinguished probability space, $(\Omega, \Sigma, P)$, associated with every model of a discrete probabalistic scenario, namely a probability space such that for every $\omega \in \Omega$, we have

$$P(\left \{ \omega \right \})=\frac{1}{|\Omega|},$$

and does this probability space have a name to distinguish it from all of the other probability spaces that might be used in talking about this same scenario? Is the name sample space reserved for the set of outcomes associated with this probability space, or is it also applied to the set of outcomes associated with any probability space.

In Permuter's example, could we just as well not mention any random variable and say: we have this probability space, $\Omega = \left \{ 1,2,3,4,5,6 \right \}$; its events are the power set of $\Omega$; its measure is the probability measure specified as above. And then define entropy directly as a property of a probability measure, regardless of what random variables - if any - it may be the distribution of?

Third, you use a LaTeX symbol \Rset with doesn't display in my browser (Firefox). Does this denote the same as $\mathbb{R}$, the set of real numbers? Does this mean that a random variable needn't be onto? If it's the case that the range of a random variable can be a proper subset of its codomain, does such a random variable fail to induce a probability measure on all of its codomain?

 

[pwsnafu]

First, just to avoid confusion, I gather random variable has two senses: (1) a measurable function; (2) a real-valued measurable function; and I think you're using it in sense 2.

Yes, because this is the most cases you only need case number 2. The most general case is the domain being a probability space and the codomain being a measurable space. In practice, even if the domain is not the reals, it will be http://en.wikipedia.org/wiki/Polish_space" , and the "measurable set" will be the intuitive sets.

Second, is there some special, distinguished probability space, $(\Omega, \Sigma, P)$, associated with every model of a discrete probabalistic scenario, namely a probability space such that for every $\omega \in \Omega$, we have

$$P(\left \{ \omega \right \})=\frac{1}{|\Omega|},$$

and does this probability space have a name to distinguish it from all of the other probability spaces that might be used in talking about this same scenario?

Unless you specify something else, this is taken as default. Uniform distribution.

In Permuter's example, could we just as well not mention any random variable and say: we have this probability space, $\Omega = \left \{ 1,2,3,4,5,6 \right \}$; its events are the power set of $\Omega$; its measure is the probability measure specified as above. And then define entropy directly as a property of a probability measure, regardless of what random variables - if any - it may be the distribution of?

Haven't read the PDF. But it's probably not a good idea to ignore random variables all together, considering it's so common in the literature.

Third, you use a LaTeX symbol \Rset with doesn't display in my browser (Firefox). Does this denote the same as $\mathbb{R}$, the set of real numbers? Does this mean that a random variable needn't be onto? If it's the case that the range of a random variable can be a proper subset of its codomain, does such a random variable fail to induce a probability measure on all of its codomain?

My bad, \Rset is a LaTeX macro I use at work. And yes it means the reals.

As for the last question the answer is no. Because a random variable is measurable, the http://en.wikipedia.org/wiki/Pushforward_measure" always exists. Let's continue the example above.

First I need a probability measure on Ω, let's take the uniform and call it μ. The random variable has range $\{-1,0,1\}$. I want a measure on the reals induced by μ. Let's call this ν.

Now let's calculate
$$\nu(\{1\}) = \mu(X^{-1}(\{1\}))=\mu(\{+_1\}\cup\{+_2\}) = \mu(\{+_1\})+\mu(\{+_2\})= \frac16+\frac16=\frac13$$
Similar calculation gives $\nu(0)=\nu(-1)=\frac13$ as well.

To see that ν is defined elsewhere observe
$$\nu(\{2\}) = \mu(X^{-1}(\{2\})) = \mu(\emptyset) = 0$$
This is true for any set which does not contain -1,0, or 1. It is just a trivial extension.

Lastly, we have distinguished μ and ν. But that was because I chose to distinguish the two pluses right at the start. It I didn't do this, we wouldn't need to distinguish the measures either.

 

[Rasalhague]

To see that ν is defined elsewhere observe
$$\nu(\{2\}) = \mu(X^{-1}(\{2\})) = \mu(\emptyset) = 0$$
This is true for any set which does not contain -1,0, or 1. It is just a trivial extension.

Ah yes, of course! I see now. Thanks for explaining it.