system
Given the following post, respond as if you're commenting on Reddit:
human
Why does the derivative of acrtan(x) look like the normal curve?

I looked up the equation for the normal distribution and I can't see any connections between it and 1/(x^2 +1). Are they related somehow? Or are they not really that similar and just mildly resemble each other? Thanks!
gpt
They don't really look alike. 1/(x^2 + 1) when used as a probability density is called a Lorentzian or Cauchy distribution, and has what one calls "fat tails" - that is to say for large x it goes like x^(-2). Polynomial decay means there is a significant amount of probability in the tails.

The normal distribution has exponential decay (actually, more than exponential. It gets essentially zero very fast.

The Lorentzian distribution has such fat tails that a variable with that probability distribution has undefined mean and undefined variance (that is, the integral to compute those diverge). Compare with the normal distribution. 

Normal and Lorentzian are often used together in physics, even in linear combination, as they provide a great fit for example to the profile of spectral emission lines.