Cauchy-class {distr} | R Documentation |

## Class "Cauchy"

### Description

The Cauchy distribution with location `l`

, by default `=0`

, and scale `s`

, by default `=1`

,has
density

```
f(x) = \frac{1}{\pi s}
\left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}%
```

for all `x`

.
C.f. `rcauchy`

### Objects from the Class

Objects can be created by calls of the form `Cauchy(location, scale)`

.
This object is a Cauchy distribution.

### Slots

`img`

Object of class

`"Reals"`

: The domain of this distribution has got dimension 1 and the name "Real Space".`param`

Object of class

`"CauchyParameter"`

: the parameter of this distribution (location and scale), declared at its instantiation`r`

Object of class

`"function"`

: generates random numbers (calls function`rcauchy`

)`d`

Object of class

`"function"`

: density function (calls function`dcauchy`

)`p`

Object of class

`"function"`

: cumulative function (calls function`pcauchy`

)`q`

Object of class

`"function"`

: inverse of the cumulative function (calls function`qcauchy`

)`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

`Symmetry`

object of class

`"DistributionSymmetry"`

; used internally to avoid unnecessary calculations.

### Extends

Class `"AbscontDistribution"`

, directly.

Class `"UnivariateDistribution"`

, by class `"AbscontDistribution"`

.

Class `"Distribution"`

, by class `"AbscontDistribution"`

.

### Is-Relations

By means of `setIs`

, R “knows” that a distribution object `obj`

of class `"Cauchy"`

with location 0 and scale 1 also is
a T distribution with parameters `df = 1, ncp = 0`

.

### Methods

- initialize
`signature(.Object = "Cauchy")`

: initialize method- location
`signature(object = "Cauchy")`

: returns the slot`location`

of the parameter of the distribution- location<-
`signature(object = "Cauchy")`

: modifies the slot`location`

of the parameter of the distribution- scale
`signature(object = "Cauchy")`

: returns the slot`scale`

of the parameter of the distribution- scale<-
`signature(object = "Cauchy")`

: modifies the slot`scale`

of the parameter of the distribution- +
`signature(e1 = "Cauchy", e2 = "Cauchy")`

: For the Cauchy distribution the exact convolution formula is implemented thereby improving the general numerical approximation.- *
`signature(e1 = "Cauchy", e2 = "numeric")`

- +
`signature(e1 = "Cauchy", e2 = "numeric")`

: For the Cauchy location scale family we use its closedness under affine linear transformations.

further arithmetic methods see operators-methods

### Author(s)

Thomas Stabla statho3@web.de,

Florian Camphausen fcampi@gmx.de,

Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de,

Matthias Kohl Matthias.Kohl@stamats.de

### See Also

`CauchyParameter-class`

`AbscontDistribution-class`

`Reals-class`

`rcauchy`

### Examples

```
C <- Cauchy(location = 1, scale = 1) # C is a Cauchy distribution with location=1 and scale=1.
r(C)(1) # one random number generated from this distribution, e.g. 4.104603
d(C)(1) # Density of this distribution is 0.3183099 for x=1.
p(C)(1) # Probability that x<1 is 0.5.
q(C)(.1) # Probability that x<-2.077684 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(C) # location of this distribution is 1.
location(C) <- 2 # location of this distribution is now 2.
is(C,"Td") # no
C0 <- Cauchy() # standard, i.e. location = 0, scale = 1
is(C0,"Td") # yes
as(C0,"Td")
```

*distr*version 2.9.3 Index]