Function looks like a distribution curve

Question

I was playing around with Desmos and noticed that the graph of $y=(x^2+1)^{-1}$ looks like a distribution. I know that distributions like the normal distribution follow the form of $a^{-x^2}$. Are these functions related at all? Does the function I've mentioned have any use? Thanks.

Answer 1

The density function $$f(x)=\frac1{\pi(1+x^2)}$$ defines the Cauchy distribution. It's a great example in probability theory. It has no mean, no variance, and does not satisfy the central limit theorem!