Forgetting about the Underlying Probability Space

Question

I have read that when dealing with random variables, one often forgets about the underlying probability space (Wikipeida). What is a good example (or a couple) of when one does this? I figure that there are two ways to go about this:

(1) a stand-alone random variable where the probability space is too complicated or

(2) two random variables on rather different probability spaces that have the same distribution, which motivates the study of the distributions themselves.

By a good example, I mean one that would motivate the study of distributions without regard to the underlying probability space. I am not looking for an example that could just so happen be studied with distributions, but an example that shows that it is easier to study the random variable with respect to the distribution rather than the underlying probability space.

Answer 1

I guess a simple example is if you have a random variable like $X$ which has a uniform probability distribution from 0 to 1 on the real x axis. The value of $X$ is just a real number from 0 to 1, but the underlying probability space (i.e. the random outcome to which this real number is associated) can't be easily thought of as some kind of roll of a very-many-sided die. I hope this helps.

Answer 2

Let $(\Omega, \mathcal A, P)$ be a probability space, and $(\Omega', \mathcal A')$ another measurable space. Let $X$ be a measurable function, \begin{align*} X : \Omega \to \Omega'. \end{align*} We call $X$ a random variable. The distribution of $X$ is the map $F_X$, defined by \begin{align*} F_X := P\circ X & : \mathcal A' \to [0,1] \\ & : E \mapsto P(X^{-1}(E)). \end{align*} So I suppose what the author of that wikipedia section could mean is that the distribution ignores or forgets about the underlying prob space because it is only indirectly related to it through the behaviour of $X$.

Here's an example.

Roll a dice, and for a result $x$ you eat $|3-x|$ chocolates. The random variable $X$ representing the random number of chocolates you will eat is given by \begin{align*} X & : \{1,2,3,4,5,6\} \to \{0,1,2,3\} \\ & : x \mapsto |3-x|. \end{align*} We can easily calculate the distribution function everywhere, and it doesn't bear much resemblance to the underlying experiment of rolling the dice. The more important part is the action of the random variable $X$. $$ F_X(0) = 1/6, \quad F_X(1) = 1/3, \quad F_X(2) = 1/3, \quad F_X(3) = 1/6. $$ Note: I've slightly abused notation and written $F_X(x)$ for $F_X(\{x\})$. I'm not sure I personally agree with the wikipedia statement but I hope this helps.

Answer 3

First, I believe that when we are dealing with random variables we are forgetting about the underlying sample space, not the probability space. We are forgetting about probability space much earlier, when we start dealing with probability theory itself, not with measure-theoretic grounds of probability theory.

When we are dealing with random variables, sample space is just a theoretic concept. We think that sample space is "always there", but we don't care about specifying the sample space. All we need to know is (joint) distribution of random variable(s); sample space is simply not needed.

You can find many examples of forgetting about the sample space in statistics. Statisticians usually completely forget even about existence of underlying sample space, even though in many cases it is not hard to specify a sample space. As as example consider simple hypothesis testing problem: I toss a coin 100 times and observe 60 Heads; should I decide that the coin is unfair at significance level $5\%$? Evidently the underlying sample space consists of 100 elements, the outcomes of individual tosses, but you hardly find it mentioned in statistical books. All statisticians care are random variables associated with outcomes.