Cauchy Distribution Calculator

Enter a location x0 and scale gamma to get the Cauchy distribution density, cumulative probability (lower tail), and upper tail probability, with charts.

Input

For a Cauchy distribution with location x0 and scale gamma, compute the density, cumulative distribution (lower probability), and upper tail probability at the point x.

Evaluation point x

Enter the value x at which to evaluate the probability.

Location x0

The center of the distribution (median and mode). Any real number.

Scale gamma

A positive value controlling the spread. The full width at half maximum is 2 gamma.

Result

Lower probability F(x) up to x = 1

0.75

Upper probability 1 - F(x)

0.25

Density f(x)

0.15915494

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The Cauchy density is f(x) = 1 / (pi gamma (1 + ((x - x0) / gamma) squared)). Here x0 is the location parameter giving the center (the mode and median), and gamma is the scale parameter controlling the spread.
The cumulative distribution (lower probability) is F(x) = 1/2 + arctan((x - x0) / gamma) / pi. At x = x0 the value is F = 0.5. The upper tail probability is 1 - F(x).
Because the Cauchy distribution has very heavy tails, both its mean and variance are undefined (the defining integrals diverge). Use the median x0 as the representative center instead of a mean.
The scale parameter gamma must be positive. In physics this is the Lorentzian distribution, and its full width at half maximum equals 2 gamma.

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