The Cauchy distribution is a continuous probability distribution notable for its heavy tails: it has no finite mean or variance. It arises as the ratio of two independent standard normal random variables and as the Student's t-Distribution with one degree of freedom.
A random variable \(X\) follows a Cauchy distribution with location parameter \(x_0 \in \mathbb{R}\) and scale parameter \(\gamma > 0\) if
\[ X \sim \text{Cauchy}(x_0, \gamma). \]
The standard Cauchy distribution is the case \(x_0 = 0\), \(\gamma = 1\).
The probability density function is
\[ f(x) = \frac{1}{\pi\gamma \left[1 + \left(\dfrac{x - x_0}{\gamma}\right)^2\right]}, \quad x \in \mathbb{R}. \]
Interpretation: The density is symmetric and bell-shaped about \(x_0\), but its tails decay only like \(x^{-2}\), so extreme values occur far more often than for the Normal Distribution.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.stats import cauchy, norm
sns.set_style('whitegrid')
sns.set_palette('Set2')
x = np.linspace(-6, 6, 500)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(x, cauchy.pdf(x), lw=2, label='Cauchy(0, 1)')
ax.plot(x, norm.pdf(x), lw=2, ls='--',
label=r'$\mathcal{N}(0, 1)$')
ax.set_xlabel('$x$')
ax.set_ylabel('$f(x)$')
ax.set_title('Standard Cauchy vs. standard normal')
ax.legend()
plt.show()
The cumulative distribution function is
\[ F(x) = \frac{1}{\pi} \arctan\!\left(\frac{x - x_0}{\gamma}\right) + \frac{1}{2}. \]
For \(X \sim \text{Cauchy}(x_0, \gamma)\), the expectation \(\mathbb{E}[X]\) does not exist, and consequently neither the variance nor the moment generating function exists.
Proof. For the standard Cauchy, \(\mathbb{E}[|X|] = \int_{-\infty}^\infty \frac{|x|}{\pi(1 + x^2)} \, dx\) diverges, since the integrand behaves like \(1/(\pi|x|)\) for large \(|x|\). Hence \(\mathbb{E}[X]\) is undefined and no higher moments exist.
The location \(x_0\) is instead the median and mode, and \(\gamma\) is the half-width at half-maximum.
Although the MGF does not exist, the characteristic function of \(X \sim \text{Cauchy}(x_0, \gamma)\) is well defined:
\[ \varphi_X(t) = \exp\!\left(i x_0 t - \gamma |t|\right). \]
A consequence is that the sample mean of \(n\) i.i.d. standard Cauchy variables is again standard Cauchy — averaging does not reduce dispersion, in stark contrast to the Law of Large Numbers.
The ratio of two independent centred normal measurements — for example the tangent of a uniformly random angle — follows a Cauchy distribution.
In physics, the Cauchy (Lorentzian) distribution describes the shape of spectral lines and the response of a driven resonant system.
Because it lacks a mean, the Cauchy distribution is a standard stress test for estimators and for the assumptions behind the Central Limit Theorem.