# How common are probability distributions with a finite variance?

It's always very surprising to learn that some of the entities one has been assiduously studying actually represent negligibly tiny minorities (e.g. continuous functions vis-à-vis all functions)...

Now, the Central Limit Theorem, for one, holds only for probability distributions with a finite variance.

How common are such distributions in the space of all probability distributions?

More formally, let $U$ be the set of all probability distributions on $\mathbb{R}$ (say), and $F \subset U$ be the subset of those probability distributions that have a finite variance. I expect that, of the cardinalities $|F|$ and $|U\setminus F|$, one will be a strictly larger infinity than the other, but I have no intuition as to which.

(I guess this is a "meta-measure theory" question.)

• What's stopping you there -- why not asking about those with an expectation? – Clement C. Aug 22 '17 at 11:58
• @ClementC: the motivation for my question was wanting to gauge the CLT's generality, but you bring up an excellent point. – kjo Aug 22 '17 at 12:01
• A small note, but you'll want to consider probability distributions up to a.e. equality at most, because there are $\mathfrak c$-sized subsets of measure zero on which you can choose any function value you want, so that any probability distribution already has $\mathfrak{c}^\mathfrak{c}$-many probability distributions a.e. equal to it. – Mees de Vries Aug 22 '17 at 12:08
• @kjo a different, but related point: $L_2$ is dense in $L_1$. – Clement C. Aug 22 '17 at 12:10

I presume that you speak about distributions on the Borel $\sigma$-algebra $\mathcal B(\mathbb R)$. In terms of cardinality, there are quite few of them, namely, $\mathfrak{c} = |\mathbb{R}|$. Indeed, each probability distribution $\mathcal B(\mathbb R)$ is uniquely determined by its values on the intervals $(-\infty,q)$ with $q\in\mathbb{Q}$. Therefore, the number of probability measures does not exceed the number of sequences of real numbers, which is $\mathfrak{c}$. Since, obviously, there are both $\mathfrak{c}$ distributions with finite and infinite variance, so your guess is wrong.
• @MeesdeVries, yes, there are plenty. Take $\arctan$ of any infinite variance variable to get a finite variance one. But this is a good point. – zhoraster Aug 22 '17 at 12:30
• @zhoraster It's early here, so I may very well spout nonsense. But in your latest comment: if $X$ has an infinite variance due to non-integrability at $0$, wouldn't $\arctan X$ have exactly the same problem? – Clement C. Aug 22 '17 at 12:49