Cauchy Probability Density

The Cauchy distribution describes ratios of normal variables and exhibits extremely heavy tails. Its probability density function is

$f\left(x\right)=\frac{1}{\pi \gamma \left(1+{\frac{x-x_0}{\gamma }}^2\right)}$

Unlike the normal distribution, the Cauchy distribution lacks finite mean or variance. That means sample averages do not converge to a typical value even for very large samples. Because the density decays only as $\mathrm{|x|}^\{-2\}$, extremely large observations remain probable. This property makes the Cauchy distribution an important counterexample in probability theory and underscores the importance of checking assumptions behind statistical estimators.

Distribution Function

The cumulative distribution function (CDF) has a closed form involving the arctangent:

$F\left(x\right)=\frac{1}{\pi }\arctan \left(\frac{x-x_0}{\gamma }\right)+\frac{1}{2}$

This step function shows that half of the probability mass lies to the left of the location parameter $x_0$. The heavy tails mean the distribution is more spread out than a normal distribution with comparable scale.

The Cauchy distribution arises in various contexts from resonance behavior in physics to Bayesian statistics. Its long tails sometimes model phenomena with rare but extreme deviations. Because its moments do not exist, common methods such as least squares can behave poorly under Cauchy-like noise. In such settings, robust estimators like the median or interquartile range provide more reliable summaries.

When analyzing data, it can be valuable to visualize how quickly probabilities accumulate in the Cauchy distribution. This calculator lets you experiment with different locations $x_0$, scales $\gamma$, and evaluation points $x$. Observe how the height of the density function decreases slowly compared with the normal distribution, and how the CDF transitions gradually from 0 to 1. Such experimentation builds intuition for heavy-tailed behavior.