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Cauchy Distribution Percent Point Calculator

Find the percent point (quantile) of a Cauchy distribution in closed form from probability p, location x0, and scale gamma. Choose lower or upper tail and view the density and CDF curves.

Input

Enter the probability p, location parameter x0, and scale parameter gamma to get the percent point (quantile) of the Cauchy distribution in closed form.

Enter a value greater than 0 and less than 1.

Tail

Choose whether p is treated as a lower tail or upper tail probability.

The center of the distribution (its median).

A positive value for the spread of the tails. It equals the half width at half maximum.

Result

Percent point x for lower tail probability p = 0.95

6.31375151

Location x0

0

Scale gamma

1

Median

0

Full width at half max 2 gamma

2

Lower probability F(x)

0.95

Upper probability 1 minus F(x)

0.05

Density f(x)

0.0077896

Mean and variance

Undefined

Probability density function f(x)

Cumulative distribution function F(x)

How it works

  • The Cauchy cumulative distribution is F(x) = 1/2 + (1/pi) arctan((x minus x0) / gamma).
  • The lower tail percent point has the closed form x = x0 + gamma tan(pi(p minus 1/2)).
  • For an upper tail probability, p is replaced by 1 minus p before computing.
  • The scale parameter gamma must be positive, and the full width at half maximum is 2 gamma.
  • The Cauchy distribution has heavy tails, so its mean and variance are undefined. The median x0 serves as the central value.

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