1 min read

Random variables and forgetting formal structure

A strange thing about statistics is that it is full of functions – mapping relationships – where we don't care what the mapping relationship is.

The most basic objects defined in statistical theory are random variables. These are defined as functions from some sample space \(\Omega\) to the set of real numbers \(\mathbb{R}\).

What is strange about random variables is that they are functions, but we don't care about the mapping relationship at all. It is totally irrelevant to statistical practice. It is not at all of interest except in a heuristic way. Most of the time we forget entirely about these things being defined as functions.

Heuristically this this seems pretty reasonable, but it is sort of strange. The sample space is there to make the math work but (from what I understand) it doesn't affect the math that much. Still, it is weird to think about.