# Random Variables - Distinction between their Domain and Image spaces.

A random variable is as a function from a sample space $$\Omega$$ (set of outcomes of an experiment) to some measurable space $$E$$.

$$X:\Omega \rightarrow E$$

Wikipedia says the $$\textbf{values}$$ of a random variable $$X$$ are the possible outcomes of an experiment. However, I thought the sample space $$\Omega$$ is the set of possible outcomes of an experiment, so it's really the $$\textbf{arguments}$$ of $$X$$. Which one is it, the values of $$X$$ or the arguments of $$X$$ that are the possible outcomes? It has to be the arguments according to this definition, but all my statistics textbooks call them the values.

If $$\Omega$$ is the set of outcomes, then what would you call $$E$$? For example, suppose I roll a die, the possible outcomes are $$\Omega=\{1,2,3,4,5,6\}$$

So $$X(1)=e_1 \in E$$, what meaning would you assign to this $$e_1$$?

Because this doesn't make sense either, here is my alternative interpretation: In this example, E is really the set of values $$\{1,2,3,4,5,6\}$$, and $$\Omega$$ is the set of literal, physical outcomes, like the face of the die with the two dots coming up, or the face of the die with the six dots coming face up. This way, $$X$$ is really a function that assigns a real number to each outcome, and hopefully that real number is some sort of consistent interpretation, but $$X$$ is really an interpretation function that 'quantifies' the experiment's outcomes. Does this make more sense?

Thanks!

I think you are misreading the formal definition of a random variable. The domain is the measure space of possible outcomes. The codomain is the real numbers: the possible values of a measurement that depends on the outcome. So for pair of dice the set of outcomes is the obvious $$36$$ element set. If the dice are fair then the probability measure on that finite set is the uniform one. The sum of a roll is an example of a random variable.
• Does that mean that there is no random variable to be discussed until we perform a measurement on the set of outcomes? In a way that this measurement results in values in a measurable space $E$. – Mike Nov 16 '18 at 15:53
• Any real valued function on the sample space is a random variable and a "measurement". You don't have to "perform" anything. (This is not quantum mechanics, where probability enters in a more subtle way.) For two dice you can think of the sum, or the average, or the number of threes. These are all random variables. Confusingly, "measure" does come into the discussion as the probability of a set of outcomes (a subset of the sample space). So the measure of the set in which at least one of the (fair) dice shows six is $11/36$. – Ethan Bolker Nov 16 '18 at 15:57