SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class InverseGaussianDist

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

`public class InverseGaussianDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the inverse Gaussian distribution with location parameter μ > 0 and scale parameter λ > 0. Its density is

f (x) = (λ/ (2πx^3))1/2e-λ(x-μ)2/(2μ2x),         for x > 0.

The distribution function is given by

F(x) = Φ((λ/x)1/2(x/μ -1)) + e2λ/μΦ(- (λ/x)1/2(x/μ + 1)),

where Φ is the standard normal distribution function.

The non-static versions of the methods cdf, barF, and inverseF call the static version of the same name.

Field Summary

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

Constructor Summary
```InverseGaussianDist(double mu, double lambda)```
Constructs the inverse Gaussian distribution with parameters μ and λ.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double mu, double lambda, double x)```
Computes the complementary distribution function of the inverse gaussian distribution with parameters μ and λ, evaluated at x.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double mu, double lambda, double x)```
Computes the distribution function of the inverse gaussian distribution with parameters μ and λ, evaluated at x.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double mu, double lambda, double x)```
Computes the density function for the inverse gaussian distribution with parameters μ and λ, evaluated at x.
`static InverseGaussianDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of an inverse gaussian distribution with parameters μ and λ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getLambda()`
Returns the parameter λ of this object.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double mu, double lambda)```
Returns the mean E[X] = μ of the inverse gaussian distribution with parameters μ and λ.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (μ, λ) of the inverse gaussian distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMu()`
Returns the parameter μ of this object.
` double[]` `getParams()`
Return a table containing the parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double mu, double lambda)```
Computes and returns the standard deviation of the inverse gaussian distribution with parameters μ and λ.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double mu, double lambda)```
Computes and returns the variance Var[X] = μ3/λ of the inverse gaussian distribution with parameters μ and λ.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double mu, double lambda, double u)```
Computes the inverse of the inverse gaussian distribution with parameters μ and λ.
` void` ```setParams(double mu, double lambda)```
Sets the parameters μ and λ of 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

### InverseGaussianDist

```public InverseGaussianDist(double mu,
double lambda)```
Constructs the inverse Gaussian distribution with parameters μ and λ.

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 lambda,
double x)```
Computes the density function for the inverse gaussian distribution with parameters μ and λ, evaluated at x.

### cdf

```public static double cdf(double mu,
double lambda,
double x)```
Computes the distribution function of the inverse gaussian distribution with parameters μ and λ, evaluated at x.

### barF

```public static double barF(double mu,
double lambda,
double x)```
Computes the complementary distribution function of the inverse gaussian distribution with parameters μ and λ, evaluated at x.

### inverseF

```public static double inverseF(double mu,
double lambda,
double u)```
Computes the inverse of the inverse gaussian distribution with parameters μ and λ.

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (μ, λ) of the inverse gaussian 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, λ].

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(λ)]

### getInstanceFromMLE

```public static InverseGaussianDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of an inverse gaussian distribution with parameters μ and λ 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 lambda)```
Returns the mean E[X] = μ of the inverse gaussian distribution with parameters μ and λ.

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

### getVariance

```public static double getVariance(double mu,
double lambda)```
Computes and returns the variance Var[X] = μ3/λ of the inverse gaussian distribution with parameters μ and λ.

Returns:
the variance of the inverse gaussian distribution Var[X] = μ3/λ

### getStandardDeviation

```public static double getStandardDeviation(double mu,
double lambda)```
Computes and returns the standard deviation of the inverse gaussian distribution with parameters μ and λ.

Returns:
the standard deviation of the inverse gaussian distribution

### getLambda

`public double getLambda()`
Returns the parameter λ of this object.

### getMu

`public double getMu()`
Returns the parameter μ of this object.

### setParams

```public void setParams(double mu,
double lambda)```
Sets the parameters μ and λ of this object.

### getParams

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

### 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.