Cauchy distribution

### 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

cauchy(2.2,1.5,0.8)

0.113412073462 |  |  |  |  |  |  |  |  |  |

cauchy(0.3)

0.292027418517 |  |  |  |  |  |  |  |  |  |

##### 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

cauchy_cdf(2,3,1.4)

0.437167041811 |  |  |  |  |  |  |  |  |  |

cauchy_cdf(1.4)

0.802568456711 |  |  |  |  |  |  |  |  |  |

cauchy_cdf(2,3,-1.9,1.4)

0.228452641651 |  |  |  |  |  |  |  |  |  |

##### 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

cauchy_icdf(2,3,0.23)

−1.40283204777 |  |  |  |  |  |  |  |  |  |