SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class LaplaceDist

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

`public class LaplaceDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the Laplace distribution. It has location parameter μ and scale parameter β > 0. The density function is given by

f (x) = e-| x-μ|/β/(2β)         for - ∞ < x < ∞.

The distribution function is

 F(x) = (1/2)e(x-μ)/β if x <= μ, F(x) = 1 - (1/2)e(μ-x)/β otherwise,

and its inverse is

 F-1(u) = βlog(2u) + μ if 0 <= u <= 1/2, F-1(u) = μ - βlog(2(1 - u)) otherwise.

Field Summary

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

Constructor Summary
`LaplaceDist()`
Constructs a LaplaceDist object with default parameters μ = 0 and β = 1.
```LaplaceDist(double mu, double beta)```
Constructs a LaplaceDist object with parameters μ = mu and β = beta.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double mu, double beta, double x)```
Computes the Laplace complementary distribution function.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double mu, double beta, double x)```
Computes the Laplace distribution function.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double mu, double beta, double x)```
Computes the Laplace density function.
` double` `getBeta()`
Returns the parameter β.
`static LaplaceDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of a Laplace distribution with parameters μ 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 mu, double beta)```
Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (μ, β) of the Laplace distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMu()`
Returns the parameter μ.
` double[]` `getParams()`
Return a table containing the parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double mu, double beta)```
Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double mu, double beta)```
Computes and returns the variance Var[X] = 2β2 of the Laplace distribution with parameters μ and β.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double mu, double beta, double u)```
Computes the inverse Laplace distribution function.
` 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

### LaplaceDist

`public LaplaceDist()`
Constructs a LaplaceDist object with default parameters μ = 0 and β = 1.

### LaplaceDist

```public LaplaceDist(double mu,
double beta)```
Constructs a LaplaceDist object with parameters μ = mu 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 mu,
double beta,
double x)```
Computes the Laplace density function.

### cdf

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

### barF

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

### inverseF

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

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (μ, β) of the Laplace 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: [μ, β].

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

### getInstanceFromMLE

```public static LaplaceDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of a Laplace distribution with parameters μ 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 mu,
double beta)```
Computes and returns the mean E[X] = μ of the Laplace distribution with parameters μ and β.

Returns:
the mean of the Laplace distribution E[X] = μ

### getVariance

```public static double getVariance(double mu,
double beta)```
Computes and returns the variance Var[X] = 2β2 of the Laplace distribution with parameters μ and β.

Returns:
the variance of the Laplace distribution Var[X] = 2β2

### getStandardDeviation

```public static double getStandardDeviation(double mu,
double beta)```
Computes and returns the standard deviation of the Laplace distribution with parameters μ and β.

Returns:
the standard deviation of the Laplace distribution

### getMu

`public double getMu()`
Returns the parameter μ.

### getBeta

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

### getParams

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