Clarifying the role of a random variable

Question

A random variable $X$ is a mapping from a probability space $(\Omega, \mathcal{F}, \mathcal{P})$ to a measurable space $(E, \mathcal{E})$.

A sample space $\Omega$ usually include non-mathematical objects (e.g. the names of a bunch of people). The sigma algebra $\mathcal{F}$ contains the subsets of $\Omega$ we can assign probabilities to.

A random variable maps objects in the sample space to a measurable space. So, to me it seems as if a random variable in layman terms really is just a way to be able to described/translate the non-mathematical objects in a sample space to corresponding mathematical objects (e.g. numbers) in some measurable space.

So, for example: we can describe the probability distribution of the non-mathematical objects, or, we can introduce a random variable and describe the (same) probability distribution of the random variable. Right?

Is this a correct way of thinking about it? Am I missing something?

Answer 1

To me it seems as if a random variable in layman terms really is just a way to be able to described/translate the non-mathematical objects in a sample space to corresponding mathematical objects (e.g. numbers) in some measurable space.

Not exactly. A random variable is not there to describe or translate. In a lot of cases it will be losing information (which a translation is not supposed to do).

As you have understood, the framework is that you have a random experiment, with an output that could be literally anything depending on the context, and then to each output you associate a number. Now the reason for doing this also depends on context, but one great example is that of a gambling game:

You roll a die / Pick a card from a deck / Run any random experiment, and depending on the outcome I will give you some amount of money, or you will give me some amount of money.

The amount of money, here, is the random variable.

Another context where random variables occur often is when they are actually the main object of study. For instance, if I choose a random person and compute their weight, or height, or amount of money they own, the random experiment (choosing someone at random) is not what I'm interested with, I'm interested with the distribution of my random variable.

Answer 2

The value added by having an explicit random variable instead of just its probability distribution seems, you suggest, small. I think you are largely right in the case of a single random variable, where one might as well study the distribution function by itself. Such as in the case of the De Moivre-Laplace theorem, viewed as a numerical approximation to the binomial distribution function.

But if the object of study involves many, or even infinitely many random variables simultaneously, as with the strong law of large numbers or any of the properties of the Brownian motion, it seems hard to avoid the standard $(\Omega,\mathcal F,P)$ apparatus.

This is reflected in the following naive and oversimplified historical summary: Mathematical probability theory chugged along nicely as a branch of analysis studying properties of distribution functions until (say) about 1900, when the modern formulation took hold. In the former period we have De Moivre, Laplace, Chebyshev, Markov, etc, studying distribution functions with the methods of then-classical analysis; in the latter, Borel, Lebesgue, Kolmogorov, Wiener, etc, measure-theoretically.