If your eye isn’t quite trained, you might have looked at the above picture and thought, “Oh, a family of normal / Gaussian distributions!” And… you would be wrong. The distributions above are various forms of the **Cauchy distribution** which has a wider peak and fatter tails than the normal distribution.

External image

The interesting about the Cauchy distribution though is that although it is a bell-shaped curve its *mean and variance are undefined*. Due to the equation through which the probability distribution function is defined, it is not possible to calculate the moments needed to get the mean or variance, nor is it possible to calculate any other finite moments. The Cauchy distribution does have many uses (the one I’m most familiar with is it being part of the solution to Laplace’s Equation on the upper half plane which is fascinating in itself), though it stands as a classic pathological example in probability and statistics.