SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class RayleighDist

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

`public class RayleighDistextends ContinuousDistribution`

This class extends the class `ContinuousDistribution` for the Rayleigh distribution with location parameter a, and scale parameter β > 0. The density function is

f (x) = (x-a)/β2 e-(x-a)2/(2β2)        for x >= a,

and f (x) = 0 for x < a. The distribution function is

F(x) = 1 - e-(x-a)2/(2β2)        for x >= a,

and the inverse distribution function is

F-1(u) = x = a + β(-2ln(1-u))1/2        for 0 <= u <= 1.

Field Summary

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

Constructor Summary
`RayleighDist(double beta)`
Constructs a RayleighDist object with parameters a = 0 and β = beta.
```RayleighDist(double a, double beta)```
Constructs a RayleighDist object with parameters a = a, and β = beta.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double beta, double x)```
Same as barF (0, beta, x).
`static double` ```barF(double a, double beta, double x)```
Computes the complementary distribution function.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double beta, double x)```
Same as cdf (0, beta, x).
`static double` ```cdf(double a, double beta, double x)```
Computes the distribution function.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double beta, double x)```
Same as density (0, beta, x).
`static double` ```density(double a, double beta, double x)```
Computes the density function.
` double` `getA()`
Returns the parameter a.
`static RayleighDist` ```getInstanceFromMLE(double[] x, int n, double a)```
Creates a new instance of a Rayleigh distribution with parameters a and hat(β).
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double a, double beta)```
Returns the mean a + β(π/2)1/2 of the Rayleigh distribution with parameters a and β.
`static double[]` ```getMLE(double[] x, int n, double a)```
Estimates the parameter β of the Rayleigh distribution using the maximum likelihood method, assuming that a is known, from the n observations x[i], i = 0, 1,…, n - 1.
` double[]` `getParams()`
Return an array containing the parameters of the current distribution in the order: [a, β].
` double` `getSigma()`
Returns the parameter β.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` `getStandardDeviation(double beta)`
Returns the standard deviation β(2 - π/2)1/2 of the Rayleigh distribution with parameter β.
` double` `getVariance()`
Returns the variance.
`static double` `getVariance(double beta)`
Returns the variance of the Rayleigh distribution with parameter β.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double beta, double u)```
Same as inverseF (0, beta, u).
`static double` ```inverseF(double a, double beta, double u)```
Computes the inverse of the distribution function.
` void` ```setParams(double a, double beta)```
Sets the parameters a 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

### RayleighDist

`public RayleighDist(double beta)`
Constructs a RayleighDist object with parameters a = 0 and β = beta.

### RayleighDist

```public RayleighDist(double a,
double beta)```
Constructs a RayleighDist object with parameters a = a, 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 a,
double beta,
double x)```
Computes the density function.

Parameters:
`a` - the location parameter
`beta` - the scale parameter
`x` - the value at which the density is evaluated
Returns:
the density function

### density

```public static double density(double beta,
double x)```
Same as density (0, beta, x).

Parameters:
`beta` - the scale parameter
`x` - the value at which the density is evaluated
Returns:
returns the density function

### cdf

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

Parameters:
`a` - the location parameter
`beta` - the scale parameter
`x` - the value at which the distribution is evaluated
Returns:
returns the distribution function

### cdf

```public static double cdf(double beta,
double x)```
Same as cdf (0, beta, x).

Parameters:
`beta` - the scale parameter
`x` - the value at which the distribution is evaluated
Returns:
returns the distribution function

### barF

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

Parameters:
`a` - the location parameter
`beta` - the scale parameter
`x` - the value at which the complementary distribution is evaluated
Returns:
returns the complementary distribution function

### barF

```public static double barF(double beta,
double x)```
Same as barF (0, beta, x).

Parameters:
`beta` - the scale parameter
`x` - the value at which the complementary distribution is evaluated
Returns:
returns the complementary distribution function

### inverseF

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

Parameters:
`a` - the location parameter
`beta` - the scale parameter
`u` - the value at which the inverse distribution is evaluated
Returns:
returns the inverse of the distribution function

### inverseF

```public static double inverseF(double beta,
double u)```
Same as inverseF (0, beta, u).

Parameters:
`beta` - the scale parameter
`u` - the value at which the inverse distribution is evaluated
Returns:
returns the inverse of the distribution function

### getMLE

```public static double[] getMLE(double[] x,
int n,
double a)```
Estimates the parameter β of the Rayleigh distribution using the maximum likelihood method, assuming that a is known, from the n observations x[i], i = 0, 1,…, n - 1. The estimate is returned in a one-element array: [hat(β)].

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

### getInstanceFromMLE

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

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

### getMean

```public static double getMean(double a,
double beta)```
Returns the mean a + β(π/2)1/2 of the Rayleigh distribution with parameters a and β.

Parameters:
`a` - the location parameter
`beta` - the scale parameter
Returns:
the mean of the Rayleigh distribution

### getVariance

`public static double getVariance(double beta)`
Returns the variance of the Rayleigh distribution with parameter β.

Parameters:
`beta` - the scale parameter
Returns:
the variance of the Rayleigh distribution

### getStandardDeviation

`public static double getStandardDeviation(double beta)`
Returns the standard deviation β(2 - π/2)1/2 of the Rayleigh distribution with parameter β.

Parameters:
`beta` - the scale parameter
Returns:
the standard deviation of the Rayleigh distribution

### getA

`public double getA()`
Returns the parameter a.

Returns:
the location parameter a

### getSigma

`public double getSigma()`
Returns the parameter β.

Returns:
the scale parameter beta

### setParams

```public void setParams(double a,
double beta)```
Sets the parameters a and β for this object.

Parameters:
`a` - the location parameter
`beta` - the scale parameter

### getParams

`public double[] getParams()`
Return an array containing the parameters of the current distribution in the order: [a, β].

Returns:
[a, 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.