The *Cauchy distribution* is a continuous probability distribution characterized by its heavy tails and a "peak" at its central value. It is often used as a counterexample in probability and statistics because it exhibits unusual properties compared to other distributions. ![[cauchy.png|center|400]] ## Probability Density Function (PDF) The [[Probability Density Function|PDF]] of a Cauchy distribution is given by: $ f(x; x_0, \gamma)= \frac{1}{\pi\gamma \left(1+ (\frac{x-x_0}{\gamma})^2\right) }$ where: - $x_0$: Location parameter (the peak of the distribution). It is not the mean, as this is undefined. - $\gamma$: Scale parameter (controls the width). Must be positive. - $\pi$: Ensures normalization to 1. When $x_0=0$ and $\gamma=1$, the distribution is referred to as the standard Cauchy distribution. ## Key Properties - *No finite moments:* The Cauchy distribution does not have finite moments (e.g. [[Expectation]], [[Variance]]) - *Symmetry:* It is symmetric about $x_0$ - *Heavy tails:* The tails decay as $1 \over x^2$, which is much slower than the exponential decay of a [[Gaussian Distribution]]. This causes extreme values to have significant weight.