Finding variance of $X$ when $X$ has pdf $f(x)=|x|^{-3}\mathbf{1}[|x|>1]$

Question

My attempt is as follows; I notice $f(x)$ is even and therefore has 0 1st moment. Also $x^2f$ is also even, so we can write;

$$Var(X)=E(X^2)=2\int_1^{\infty}x^2(x^{-3})\,dx$$

But this dosn't converge...

This must be wrong since I'm later I'm asked to use CLT on a sample.

Thanks in advance.

Answer 1

But you're not wrong; the variance really is infinite in this case. I'd have to see the exact wording of what you're later asked to do, but you might be expected to pick up on the CLT being unsuitable for this distribution. either that or the exponent $3$ is a typo, but in that case there'd need to be an overall proportionality constant on the PDF.