20.4.14  Cauchy distribution

The probability density function for the Cauchy distribution.

The cauchy (or cauchyd) command computes the probability density function for the Cauchy distribution (sometimes called the Lorentz distribution).

• cauchy takes two optional arguments and one mandatory argument:
  • Optionally, a and b, real numbers (the parameters; by default a=0 and b=1).
  • x, a real number.
• cauchy(⟨ a,b,⟩ x) returns the value of the density function at x, namely b/π/(x−a)²+b².

Examples

The cumulative distribution function for the Cauchy distribution.

The cauchy_cdf (or cauchyd_cdf) command computes the cumulative distribution function for the Cauchy distribution.

• cauchy_cdf (or cauchyd_cdf) takes three optional arguments and one mandatory argument:
  • Optionally, a and b, the parameters (by default, a=0 and b=1).
  • x, a real number.
  • Optionally, y, a real number.
• cauchy_cdf(⟨ a,b,⟩ x) returns Prob(X ≤ x) for the Cauchy distribution with parameters a and b.
• cauchy_cdf(⟨ a,b,⟩ x,y) returns Prob(x ≤ X ≤ y).

It turns out that cauchy_cdf(a,b,x)=1/2+1/πarctanx−a/b.

Examples

The inverse distribution function for the Cauchy distribution.

The cauchy_icdf (or cauchyd_icdf) command computes the inverse distribution for the Cauchy distribution.

• cauchy_icdf (or cauchyd_icdf) takes two optional arguments and one mandatory argument:
  • Optionally, a and b, parameters (by default, a=0 and b=1).
  • h, a real number between 0 and 1.
• cauchy_icdf([a,b,] h) returns the inverse distribution for the Cauchy distribution with parameters a and b; namely, the value of x for which Prob(X ≤ x)=h.

Example