Cauchy Distribution — Definition, Formula & Examples

The Cauchy distribution is a continuous probability distribution shaped like a bell curve but with much heavier tails, meaning extreme values occur far more often than with a normal distribution. Its most distinctive property is that it has no defined mean or variance.

A continuous random variable $X$ follows a Cauchy distribution with location parameter $x_0$ and scale parameter $\gamma > 0$ if its probability density function is $f(x) = \frac{1}{\pi\gamma\left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]}$ for $x \in (-\infty, \infty)$ . The standard Cauchy distribution sets $x_0 = 0$ and $\gamma = 1$ , and is equivalent to a $t$ -distribution with 1 degree of freedom.

Key Formula

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

Where:

$x_0$ = Location parameter (the peak/median of the distribution)
$\gamma$ = Scale parameter (half-width at half-maximum), must be positive
$x$ = The continuous random variable

How It Works

The location parameter

x_0

shifts the peak of the distribution left or right, similar to the mean in a normal distribution. The scale parameter

\gamma

controls the width, analogous to standard deviation. Because the tails decay so slowly (proportional to

1/x^2

rather than exponentially), the integral that would define the mean diverges. This means the sample mean of Cauchy data does not converge to a fixed value as you collect more data — the law of large numbers does not apply. The Cauchy distribution arises naturally as the ratio of two independent standard normal random variables.

Worked Example

Problem: Find the probability density of the standard Cauchy distribution at x = 1.

Set parameters: The standard Cauchy has location x₀ = 0 and scale γ = 1.

x_0 = 0, \quad \gamma = 1

Substitute into the PDF: Plug x = 1 into the formula.

f(1) = \frac{1}{\pi \cdot 1 \cdot \left[1 + \left(\frac{1 - 0}{1}\right)^2\right]} = \frac{1}{\pi(1 + 1)}

Simplify: Compute the final value.

f(1) = \frac{1}{2\pi} \approx 0.1592

Answer: The standard Cauchy PDF evaluated at x = 1 is approximately 0.1592. For comparison, the standard normal PDF at x = 1 is about 0.2420 — the Cauchy is lower near the center but much higher in the tails.

Visualization

Why It Matters

The Cauchy distribution is a key counterexample in statistics courses: it shows that not every distribution has a mean or variance, and that the central limit theorem requires finite variance. In physics, the same distribution appears as the Lorentzian or Breit-Wigner distribution, describing resonance phenomena in spectroscopy and particle physics.

Common Mistakes

Mistake: Assuming the Cauchy distribution has a mean of 0 because it is symmetric around its peak.

Correction: Symmetry alone does not guarantee that the mean exists. The integral defining the expected value diverges for the Cauchy distribution, so the mean is undefined — not zero.