Cauchy Distribution
X ~ Cauchy(x₀, γ)
Heavy-Tailed Family • Published July 22, 2026
The standard counterexample to averaging: history from Poisson's 1824 encounter with its Law-of-Large-Numbers-defying tails through Cauchy's use of it to popularize the pathology, exact ties to the already-published Student's t at one degree of freedom and to the ratio of two independent standard Normals, a mean and variance that provably do not exist (confirmed by a sample mean that never converges no matter the sample size), and a maximum-likelihood fit that beats the sample median with the textbook 8/π² asymptotic relative efficiency.
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About This Reference Sheet
The standard counterexample to averaging: history from Poisson's 1824 encounter with its Law-of-Large-Numbers-defying tails through Cauchy's use of it to popularize the pathology, exact ties to the already-published Student's t at one degree of freedom and to the ratio of two independent standard Normals, a mean and variance that provably do not exist (confirmed by a sample mean that never converges no matter the sample size), and a maximum-likelihood fit that beats the sample median with the textbook 8/π² asymptotic relative efficiency.
Support
x ∈ (-∞, ∞)
Parameters
x₀ ∈ ℝ (location), γ > 0 (scale)