Mean and variance of Cauchy-like distribution

Question

$P(x)= \frac{b}{|x|^{2.2} + 1}$

Looking at this distribution function, is there a simple integration technique (beyond residues in complex analysis) that gives us the value of b through normalization?

Secondly, for the mean and variances, I can prove that the integral converges for $xP(x)$ and diverges for $x^{2}P(x)$ through comparison with simpler series like $\frac{1}{x^{2}+a}$, would that be enough for proving existence or lack of thereof, for the mean and variance. My naive approach for estimating the mean is that the distribution is an even function and the mean exists, and it looks like a normal/cauchy shape, so the theoretical mean value has to be zero? Is that a valid argument?

Answer 1

If there is a finite mean and if the distribution is symmetric about $0$ then you are correct: the mean will be $0$. Beyond those two properties, the shape does not matter for this argument.

As for the integral, I think you may find $$b=\dfrac{2.2}{2 \pi} \sin \left(\dfrac {\pi}{2.2}\right)$$ and similarly for other exponents greater than $1$.