39  Cauchy Distribution

The Cauchy distribution is the canonical example of a distribution with no finite moments — not even the mean. Its power-law tails decay so slowly that the average of \(n\) Cauchy random variables has the same distribution as a single one, and the Central Limit Theorem fails entirely.

Formally, the random variate \(X\) defined for all of \(\mathbb{R}\), is said to have a Cauchy Distribution (i.e. \(X \sim \text{Cauchy}(x_0, \gamma)\)) with location parameter \(x_0 \in \mathbb{R}\) and scale parameter \(\gamma > 0\). The standard Cauchy has \(x_0 = 0\) and \(\gamma = 1\). In R: dcauchy(x, location = x0, scale = gamma), pcauchy, qcauchy, rcauchy.

Note: Two app modules cover this distribution — a standard Cauchy app (no parameter sliders, fixed at \(x_0 = 0\), \(\gamma = 1\)) and a location-scale app. Both are described in the R Module section.

39.1 Probability Density Function

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

The figure below shows examples of the Cauchy Probability Density Function for different parameter combinations.

Code 
par(mfrow = c(2, 2))
x <- seq(-8, 8, length = 500)

plot(x, dcauchy(x, location = 0, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(x[0] == 0, ",  ", gamma == 1)))

plot(x, dcauchy(x, location = 0, scale = 2), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(x[0] == 0, ",  ", gamma == 2)))

plot(x, dcauchy(x, location = 2, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(x[0] == 2, ",  ", gamma == 1)))

plot(x, dcauchy(x, location = 0, scale = 0.5), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "f(x)", main = expression(paste(x[0] == 0, ",  ", gamma == 0.5)))

par(mfrow = c(1, 1))
Figure 39.1: Cauchy Probability Density Function for various parameter combinations

39.2 Purpose

The Cauchy distribution is a fundamental counterexample in probability theory: it violates the conditions of the Law of Large Numbers and the Central Limit Theorem. Despite its pathological moment structure, it arises naturally as the ratio of two independent standard Normal random variables. Common appearances include:

  • Ratio of two independent Normal random variables
  • Resonance curves in physics (Lorentz/Breit-Wigner distribution)
  • Cauchy principal value integrals in complex analysis
  • Stable distribution theory: the Cauchy is a stable distribution with index \(\alpha = 1\)
  • Demonstrates the necessity of finite variance for the CLT to apply

Relation to the discrete setting. No standard discrete distribution has completely undefined moments. The closest conceptual relative is the Zeta distribution with shape parameter \(\leq 1\), for which the mean is also infinite. The Cauchy illustrates an extreme case where standard statistical summaries break down completely.

39.3 Distribution Function

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

The figure below shows the Cauchy Distribution Function for \(x_0 = 0\) and \(\gamma = 1\).

Code 
x <- seq(-8, 8, length = 500)
plot(x, pcauchy(x, location = 0, scale = 1), type = "l", lwd = 2, col = "blue",
     xlab = "x", ylab = "F(x)", main = "Cauchy Distribution Function",
     sub = expression(paste(x[0] == 0, ",  ", gamma == 1)))
Figure 39.2: Cauchy Distribution Function (location = 0, scale = 1)

39.4 Moment Generating Function

The moment generating function does not exist for any \(t \neq 0\). The characteristic function (Fourier transform) exists and is:

\[ \varphi(t) = \exp\!\bigl(i x_0 t - \gamma |t|\bigr) \]

39.5 1st Uncentered Moment

Undefined. The integral \(\int_{-\infty}^\infty x\, f(x)\, dx\) diverges (it does not converge absolutely).

39.6 2nd Uncentered Moment

Undefined. All moments \(\text{E}(|X|^n)\) are infinite for \(n \geq 1\).

39.7 3rd Uncentered Moment

Undefined.

39.8 4th Uncentered Moment

Undefined.

39.9 2nd Centered Moment

Undefined — the variance does not exist.

39.10 3rd Centered Moment

Undefined.

39.11 4th Centered Moment

Undefined.

39.12 Expected Value

Undefined. The mean does not exist in the classical sense (the integral diverges).

39.13 Variance

Undefined. The variance does not exist.

39.14 Median

\[ \text{Med}(X) = x_0 \]

The median is well defined and equals the location parameter.

39.15 Mode

\[ \text{Mo}(X) = x_0 \]

39.16 Coefficient of Skewness

Undefined — requires a finite third moment.

39.17 Coefficient of Kurtosis

Undefined — requires a finite fourth moment.

39.18 Parameter Estimation

Maximum likelihood estimation is well-defined despite the undefined moments. In R:

library(MASS)

set.seed(42)
x_obs <- rcauchy(100, location = 0, scale = 1)

fit <- fitdistr(x_obs, "cauchy")
print(fit)
    location      scale   
  -0.1163659    0.8970371 
 ( 0.1240431) ( 0.1299343)

39.19 R Module

39.19.1 RFC

Two Cauchy Distribution modules are available in RFC:

  • “Distributions / Cauchy Distribution (Standard)” — fixed at \(x_0 = 0\), \(\gamma = 1\)
  • “Distributions / Cauchy Distribution (Location-Scale)” — adjustable parameters

39.19.3 R Code

The following code demonstrates Cauchy probability calculations and the CLT failure:

x0 <- 0; gamma_par <- 1

# Density at x = 0
dcauchy(0, location = x0, scale = gamma_par)

# P(|X| < 1): probability within one "scale unit" of center
pcauchy(1, location = x0, scale = gamma_par) - pcauchy(-1, location = x0, scale = gamma_par)

# Median
cat("Median:", qcauchy(0.5, location = x0, scale = gamma_par), "\n")
[1] 0.3183099
[1] 0.5
Median: 0

39.20 Example

The Cauchy distribution demonstrates the failure of the Central Limit Theorem. The sample mean of \(n\) i.i.d. Cauchy\((0, 1)\) variates is itself Cauchy\((0, 1)\) — averaging provides no concentration of the estimate.

set.seed(42)
n <- 1000

# Sample means of n Cauchy draws — should converge to 0 if CLT held
means_200 <- replicate(200, mean(rcauchy(n)))

cat("Mean of sample means:", round(mean(means_200), 4), "\n")
cat("SD of sample means:  ", round(sd(means_200), 4), "\n")
cat("Expected SD if CLT:  ", round(1/sqrt(n), 6), "(CLT would predict ~0.032)\n")
cat("Note: sd >> 1/sqrt(n), confirming CLT does NOT apply\n")
Mean of sample means: -2.952 
SD of sample means:   39.4931 
Expected SD if CLT:   0.031623 (CLT would predict ~0.032)
Note: sd >> 1/sqrt(n), confirming CLT does NOT apply

The standard Cauchy app illustrates the distribution shape without adjustable parameters:

Interactive Shiny app (click to load).
Open in new tab

The location-scale app allows parameter adjustment. For \(x_0 = 0\), \(\gamma = 1\): P(\(|X| < 1\)) = 0.5:

Interactive Shiny app (click to load).
Open in new tab

39.21 Random Number Generator

Cauchy random variates are generated via the inverse-CDF method. Since \(F(x) = 1/2 + \arctan((x-x_0)/\gamma)/\pi\):

\[ X = x_0 + \gamma\tan\!\bigl(\pi(U - 1/2)\bigr) \sim \text{Cauchy}(x_0, \gamma) \quad \text{when } U \sim \text{U}(0,1) \]

set.seed(123)
n <- 1000; x0 <- 0; gamma_par <- 1

# Inverse-transform method
u <- runif(n)
x_inv <- x0 + gamma_par * tan(pi * (u - 0.5))

# Compare with rcauchy
x_rcauchy <- rcauchy(n, location = x0, scale = gamma_par)

cat("Inverse-transform: median =", round(median(x_inv), 4), "\n")
cat("rcauchy():         median =", round(median(x_rcauchy), 4), "\n")
cat("Theoretical median:", x0, "\n")
cat("Note: mean and sd are undefined (non-convergent)\n")
Inverse-transform: median = -0.0316 
rcauchy():         median = 0.0145 
Theoretical median: 0 
Note: mean and sd are undefined (non-convergent)
Interactive Shiny app (click to load).
Open in new tab

39.22 Property 1: Ratio of Independent Normals

If \(Z_1, Z_2 \overset{\text{i.i.d.}}{\sim} N(0, 1)\) then:

\[ \frac{Z_1}{Z_2} \sim \text{Cauchy}(0, 1) \]

This is the most natural way the Cauchy distribution arises in practice.

39.23 Property 2: Stable Distribution — Averaging Does Not Help

The Cauchy distribution is a stable distribution with index \(\alpha = 1\). If \(X_1, \ldots, X_n \overset{\text{i.i.d.}}{\sim} \text{Cauchy}(x_0, \gamma)\) then:

\[ \frac{X_1 + \cdots + X_n}{n} \sim \text{Cauchy}(x_0, \gamma) \]

Averaging \(n\) Cauchy variates produces no concentration whatsoever — the distribution of the mean is identical to the distribution of a single observation.

39.24 Property 3: Student’s t with 1 Degree of Freedom

The standard Cauchy distribution is identical to Student’s t-distribution with 1 degree of freedom:

\[ \text{Cauchy}(0, 1) = t(1) \]

See Chapter 25.