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The Cauchy distribution is a symmetric bell-shaped distribution which arises naturally as the ratio of two independent normal random variables with mean zero. The median is given by a parameter θ, while a scale parameter σ controls the spread of the distribution.

Parameter Range Description
θ −∞ < θ < ∞ Location parameter
σ σ > 0 Scale parameter

Probability Density Function

f ( x ; θ , σ ) = 1 π σ 1 1 + ( x − θ σ ) 2
F ( x ; θ , σ ) = 1 π arctan ( x − θ σ ) + 1 2

Support

− ∞ < x < ∞

Mean

Variance

Example θ σ
Let X₁ and X₂ be independent standard normal random variables. Then X₁/X₂ is a Cauchy(0, 1) random variable. 0.000 1.000
Let X₁ be a normal(0, 6) and X₂ an independent normal(0, 2) random variables. Then X₁/X₂ is a Cauchy(0, 3) random variable. 0.000 3.000
A line passes through the point (1, 2) at a uniformly random angle. The x-axis intercept of the line is a Cauchy(1, 2) random variable. 1.000 2.000

X ∼ Cauchy(θ, σ)

θ = σ =
Chart of the Cauchy distribution Chart area for displaying the Cauchy pdf, cdf, and simulation

E(X) , Var(X) , SD(X)

Although the pdf of the Cauchy is similar to that of a normal distribution in being symmetric about θ, the Cauchy distribution is heavy-tailed, with neither the mean nor the variance being defined. Because of this, the box under the graph showing the mean and standard deviation is not displayed.

Although the cdf of the Cauchy looks similar to that of a normal distribution, the Cauchy distribution is heavy-tailed, with neither the mean nor the variance being defined. Because of this, the box under the graph showing the mean and standard deviation is not displayed.

The illustration above shows a red line passing through the point (θ, σ), where the angle of the line is uniformly random. The light blue circle shows the location X of the x-axis intercept of this line, which has a Cauchy(θ, σ) distribution.

The simulation above shows a red line passing through the point (θ, σ), where the angle of the line is uniformly random. The light blue circle shows the location X of the x-axis intercept of this line, which has a Cauchy(θ, σ) distribution. The histogram accumulates the results of each simulation.

Y = ∑ Xᵢ ∼ Cauchy(Σθᵢ, Σσᵢ) 1/X ∼ Cauchy(θ/(θ² + σ²), σ/(θ² + σ²)) (X − θ)/σ ∼ Standard Cauchy

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) , Var(Y) , SD(Y)

The graph above shows the case when σ₁ = σ₂ = ··· = σₙ.

Note that by the Central Limit Theorem, the sum of i.i.d random variables with finite variance converges to a normal distribution. However, since the Cauchy distribution has infinite variance, the theorem does not apply.

Note that, since a Cauchy random variable is the ratio of two independent normal random variables with mean zero, then the reciprocal is also the ratio of two independent normal random variables with mean zero, and is therefore also Cauchy.

Note that the Cauchy distribution is a location-scale family of distributions, with location parameter θ and scale parameter σ.