Click or press any key to close help Help screen describing the interactive elements

The standard Cauchy distribution is a Cauchy distribution with location parameter 0 and scale parameter 1. It arises naturally as the ratio of two independent standard normal random variables.

Probability Density Function

f ( x ) = 1 π ( 1 + x 2 )
F ( x ) = 1 π arctan ( x ) + 1 2

Support

− ∞ < x < ∞

Mean

Variance

Example
A line passes through the origin at a uniformly chosen random angle. The slope of the line is a standard Cauchy random variable.
Let X₁ and X₂ be independent standard normal random variables. Then X₁/X₂ is a standard Cauchy random variable.
A sample of size 2 with sample mean X̄ and sample variance S² is chosen from a normal distribution with mean μ. Then √2(X̄ − μ)/S is a standard Cauchy random variable.

X ∼ Standard Cauchy

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

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

Although the pdf of the standard Cauchy is similar to that of a standard normal distribution in being symmetric about the origin, a key feature of the standard Cauchy distribution is that neither the mean nor the variance is defined. This is a consequence of having fatter tails than the standard normal distribution. Because of this, the box under the graph showing the mean and standard deviation is not displayed.

A key feature of the standard Cauchy distribution is that neither the mean nor the variance is defined. This is a consequence of the distribution having fatter tails than the standard normal distribution. 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 (0, 1), 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 standard Cauchy distribution.

The simulation above shows a red line passing through the point (0, 1), 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 standard Cauchy distribution. The histogram accumulates the results of each simulation.

Y = ∑ Xᵢ ∼ Cauchy(0, n) θ + σX ∼ Cauchy(θ, σ) 1/X ∼ Standard Cauchy

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

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

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 standard Cauchy distribution has infinite variance, the theorem does not apply.

This follows since the Cauchy distribution is a location-scale family of distributions.

Since a standard Cauchy random variable is the ratio of two independent standard normal random variables, then the reciprocal is also the ratio of two independent standard normal random variables, and is therefore also standard Cauchy.