SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class CauchyDist

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
umontreal.iro.lecuyer.probdist.CauchyDist
```
All Implemented Interfaces:
Distribution

`public class CauchyDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the Cauchy distribution with location parameter α and scale parameter β > 0. The density function is given by

f (x) = β/(π[(x - α)2 + β2]) for - ∞ < x < ∞.

The distribution function is

F(x) = 1/2 + arctan((x - α)/β)/π,                for - ∞ < x < ∞,

and its inverse is

F-1(u) = α + βtan(π(u - 1/2)).        for 0 < u < 1.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`decPrec`

Constructor Summary
`CauchyDist()`
Constructs a CauchyDist object with parameters α = 0 and β = 1.
```CauchyDist(double alpha, double beta)```
Constructs a CauchyDist object with parameters α = alpha and β = beta.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double alpha, double beta, double x)```
Computes the complementary distribution.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double alpha, double beta, double x)```
Computes the distribution function.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double alpha, double beta, double x)```
Computes the density function.
` double` `getAlpha()`
Returns the value of α for this object.
` double` `getBeta()`
Returns the value of β for this object.
`static CauchyDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of a Cauchy distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double alpha, double beta)```
Throws an exception since the mean does not exist.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (α, β) of the Cauchy distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double[]` `getParams()`
Return a table containing parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double alpha, double beta)```
Returns since the standard deviation does not exist.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double alpha, double beta)```
Returns since the variance does not exist.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double alpha, double beta, double u)```
Computes the inverse of the distribution.
` void` ```setParams(double alpha, double beta)```
Sets the value of the parameters α and β for this object.
` String` `toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, wait, wait, wait`

Constructor Detail

### CauchyDist

`public CauchyDist()`
Constructs a CauchyDist object with parameters α = 0 and β = 1.

### CauchyDist

```public CauchyDist(double alpha,
double beta)```
Constructs a CauchyDist object with parameters α = alpha and β = beta.

Method Detail

### density

`public double density(double x)`
Description copied from class: `ContinuousDistribution`
Returns f (x), the density evaluated at x.

Specified by:
`density` in class `ContinuousDistribution`
Parameters:
`x` - value at which the density is evaluated
Returns:
density function evaluated at x

### cdf

`public double cdf(double x)`
Description copied from interface: `Distribution`
Returns the distribution function F(x).

Parameters:
`x` - value at which the distribution function is evaluated
Returns:
distribution function evaluated at x

### barF

`public double barF(double x)`
Description copied from class: `ContinuousDistribution`
Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:
`barF` in interface `Distribution`
Overrides:
`barF` in class `ContinuousDistribution`
Parameters:
`x` - value at which the complementary distribution function is evaluated
Returns:
complementary distribution function evaluated at x

### inverseF

`public double inverseF(double u)`
Description copied from class: `ContinuousDistribution`
Returns the inverse distribution function x = F-1(u). Restrictions: u∈[0, 1].

Specified by:
`inverseF` in interface `Distribution`
Overrides:
`inverseF` in class `ContinuousDistribution`
Parameters:
`u` - value at which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

### getMean

`public double getMean()`
Description copied from class: `ContinuousDistribution`
Returns the mean.

Specified by:
`getMean` in interface `Distribution`
Overrides:
`getMean` in class `ContinuousDistribution`
Returns:
the mean

### getVariance

`public double getVariance()`
Description copied from class: `ContinuousDistribution`
Returns the variance.

Specified by:
`getVariance` in interface `Distribution`
Overrides:
`getVariance` in class `ContinuousDistribution`
Returns:
the variance

### getStandardDeviation

`public double getStandardDeviation()`
Description copied from class: `ContinuousDistribution`
Returns the standard deviation.

Specified by:
`getStandardDeviation` in interface `Distribution`
Overrides:
`getStandardDeviation` in class `ContinuousDistribution`
Returns:
the standard deviation

### density

```public static double density(double alpha,
double beta,
double x)```
Computes the density function.

### cdf

```public static double cdf(double alpha,
double beta,
double x)```
Computes the distribution function.

### barF

```public static double barF(double alpha,
double beta,
double x)```
Computes the complementary distribution.

### inverseF

```public static double inverseF(double alpha,
double beta,
double u)```
Computes the inverse of the distribution.

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (α, β) of the Cauchy distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [α, β].

Parameters:
`x` - the list of observations to use to evaluate parameters
`n` - the number of observations to use to evaluate parameters
Returns:
returns the parameters [ hat(α), hat(β)]

### getInstanceFromMLE

```public static CauchyDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of a Cauchy distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:
`x` - the list of observations to use to evaluate parameters
`n` - the number of observations to use to evaluate parameters

### getMean

```public static double getMean(double alpha,
double beta)```
Throws an exception since the mean does not exist.

Throws:
`UnsupportedOperationException` - the mean of the Cauchy distribution is undefined.

### getVariance

```public static double getVariance(double alpha,
double beta)```
Returns since the variance does not exist.

Returns:
.

### getStandardDeviation

```public static double getStandardDeviation(double alpha,
double beta)```
Returns since the standard deviation does not exist.

Returns:

### getAlpha

`public double getAlpha()`
Returns the value of α for this object.

### getBeta

`public double getBeta()`
Returns the value of β for this object.

### setParams

```public void setParams(double alpha,
double beta)```
Sets the value of the parameters α and β for this object.

### getParams

`public double[] getParams()`
Return a table containing parameters of the current distribution. This table is put in regular order: [α, β].

### toString

`public String toString()`
Overrides:
`toString` in class `Object`

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.