SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class ChiSquareNoncentralDist

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

`public class ChiSquareNoncentralDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the noncentral chi-square distribution with ν degrees of freedom and noncentrality parameter λ, where ν > 0 and λ > 0. Its density is

f (x) = (e-(x+λ)/2)/2(x/λ)(ν-2)/4Iν/2-1((λx)1/2)        for x > 0,

where Iν(x) is the modified Bessel function of the first kind of order ν given by

Iν(z) = ∑j=0[(z/2)ν+2j]/[j!  Γ(ν + j + 1)],

where Γ(x) is the gamma function. Notice that this distribution is more general than the chi-square distribution since its number of degrees of freedom can be any positive real number. For λ = 0 and ν a positive integer, we have the ordinary chi-square distribution.

The cumulative probability function can be written as

P[X <= x] = ∑j=0(e-λ/2(λ/2)j/j!)P[χ2ν+2j <= x],

where χ2ν+2j is the central chi-square distribution with ν + 2j degrees of freedom.

Field Summary

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

Constructor Summary
```ChiSquareNoncentralDist(double nu, double lambda)```
Constructs a noncentral chi-square distribution with ν = nu degrees of freedom and noncentrality parameter λ = lambda.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double nu, double lambda, double x)```
Computes the complementary noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double nu, double lambda, double x)```
Computes the noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double nu, double lambda, double x)```
Computes the density function for a noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.
` double` `getLambda()`
Returns the parameter λ of this object.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double nu, double lambda)```
Computes and returns the mean E[X] = ν + λ of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
` double` `getNu()`
Returns the parameter ν of this object.
` double[]` `getParams()`
Returns a table containing the parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double nu, double lambda)```
Computes and returns the standard deviation of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double nu, double lambda)```
Computes and returns the variance Var[X] = 2(ν +2λ) of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double nu, double lambda, double u)```
Computes the inverse of the noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.
` void` ```setParams(double nu, double lambda)```
Sets the parameters ν = nu and λ = lambda 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

### ChiSquareNoncentralDist

```public ChiSquareNoncentralDist(double nu,
double lambda)```
Constructs a noncentral chi-square distribution with ν = nu degrees of freedom and noncentrality parameter λ = lambda.

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 nu,
double lambda,
double x)```
Computes the density function for a noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.

### cdf

```public static double cdf(double nu,
double lambda,
double x)```
Computes the noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.

### barF

```public static double barF(double nu,
double lambda,
double x)```
Computes the complementary noncentral chi-square distribution function with ν = nu degrees of freedom and parameter λ = lambda.

### inverseF

```public static double inverseF(double nu,
double lambda,
double u)```
Computes the inverse of the noncentral chi-square distribution with ν = nu degrees of freedom and parameter λ = lambda.

### getMean

```public static double getMean(double nu,
double lambda)```
Computes and returns the mean E[X] = ν + λ of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:
the mean of the Noncentral noncentral chi-square distribution

### getVariance

```public static double getVariance(double nu,
double lambda)```
Computes and returns the variance Var[X] = 2(ν +2λ) of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:
the variance of the noncentral chi-square distribution

### getStandardDeviation

```public static double getStandardDeviation(double nu,
double lambda)```
Computes and returns the standard deviation of the noncentral chi-square distribution with parameters ν = nu and λ = lambda.

Returns:
the standard deviation of the noncentral chi-square distribution

### getNu

`public double getNu()`
Returns the parameter ν of this object.

### getLambda

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

### setParams

```public void setParams(double nu,
double lambda)```
Sets the parameters ν = nu and λ = lambda of this object.

### getParams

`public double[] getParams()`
Returns a table containing the parameters of the current distribution.

### 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.