An example of a SYMMETRIC distribution with finite mean but infinite/undefined variance?

Question

Is there a known symmetric distribution with finite 1st moment but undefined or infinite for moments>1?

Answer 1

A t-distribution with a small degrees of freedom parameter satisfies your requirements. While having one degree of freedom results in a Cauchy distribution that even lacks an expected value, $t_{\nu}$ for $\nu\in(1,2]$ has a mean of zero but infinite variance while also being symmetric about its mean.

Answer 2

Expanding on @whuber's suggestion: let $g : [0,+\infty) \rightarrow \mathbb{R}_+$ be any positive function such that

• $\int^{+\infty}_0 g(t) dt < +\infty$,

• $\int^{+\infty}_0 tg(t) dt < +\infty$ and

• $\int^{+\infty}_0 t^2 g(t) dt = +\infty$.

Let us denote $f := x\mapsto g(\vert x \vert)$. Then $\frac{f}{2\int^{+\infty}_0 g(t)dt}$ is a symmetric distribution with finite first moment, but infinite second moment.

Such $g$'s are easy to build: for example, let $\alpha>0$, and let us denote, for each $t \in [0,+\infty)$, $g(t) := 1$ if $t \leq 1$ and $g(t) := \frac{1}{t^\alpha}$ if $t>1$. Then such a $g$ satisfies the requirements if and only if $\alpha \in (2,3]$, so there are many of such $g$'s.