Cauchy distribution

Definition

Let $x_0 \in \mathbb{R}$ and $\gamma > 0$. The Cauchy distribution with location $x_0$ and scale $\gamma$ is the distribution on $\mathbb{R}$ with the probability density function $$f(x) = \frac{1}{\pi \gamma \left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)} = \frac{\gamma}{\pi \left((x - x_0)^2 + \gamma^2\right)}, \quad x \in \mathbb{R}.$$ The standard Cauchy distribution has $x_0 = 0$ and $\gamma = 1$, with density $\frac{1}{\pi(1 + x^2)}$.

Remarks

The cumulative distribution function is $F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan\left(\frac{x - x_0}{\gamma}\right)$. The median is $x_0$. The integral $\int |x| f(x) \, dx$ is infinite, so the distribution has no mean. It also has no variance. If $X_1, \dots, X_n$ are independent standard Cauchy, then their average $(X_1 + \dots + X_n)/n$ is again standard Cauchy. So the law of large numbers fails without a finite mean. The ratio $X/Y$ of two independent standard normal random variables is standard Cauchy.

Explanations

L1 human

The Cauchy distribution looks like a bell curve, but its tails are much heavier: very large values are not rare. Shine a lamp at a uniform random angle onto a wall. The point where the light hits the wall has a Cauchy distribution.