The Cauchy distribution

### 9.4.14 The Cauchy distribution

#### The probability density function for the Cauchy distribution: cauchy cauchyd

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

- cauchy takes two optionaly 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, cauchy(a,b,x) = b/(π ((x−a)2 + b2)).

Examples.

- Input:
  cauchy(2.2,1.5,0.8)
  Output:

  0.113412073462

- Input:
  cauchy(0.3)
  Output:

  0.292027418517

#### The cumulative distribution function for the Cauchy distribution: cauchy_cdf cauchyd_cdf

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 + arctan((x−a)/b)/π.

Examples.

- Input:
  cauchy_cdf(2,3,1.4)
  Output:

  0.437167041811

- Input:
  cauchy_cdf(1.4)
  Output:

  0.802568456711

- Input:
  cauchy_cdf(2,3,-1.9,1.4)
  Output:

  0.228452641651

#### The inverse distribution function for the Cauchy distribution: cauchy_icdf cauchyd_icdf

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.

cauchy_icdf(2,3,0.23)

Output:

−1.40283204777