SSJ
V. 2.6.

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
```
All Implemented Interfaces:
Distribution
Direct Known Subclasses:

`public class GammaDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the gamma distribution with shape parameter α > 0 and scale parameter λ > 0. The density is

f (x) = λαxα-1e-λx/Γ(α),        for x > 0,

where Γ is the gamma function, defined by

Γ(α) = ∫0xα-1e-xdx.

In particular, Γ(n) = (n - 1)! when n is a positive integer.

Field Summary

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

Constructor Summary
`GammaDist(double alpha)`
Constructs a GammaDist object with parameters α = alpha and λ = 1.
```GammaDist(double alpha, double lambda)```
Constructs a GammaDist object with parameters α = alpha and λ = lambda.
```GammaDist(double alpha, double lambda, int d)```
Constructs a GammaDist object with parameters α = alpha and λ = lambda, and approximations of roughly d decimal digits of precision when computing functions.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double alpha, double lambda, int d, double x)```
Computes the complementary distribution function.
`static double` ```barF(double alpha, int d, double x)```
Same as `barF` (alpha, 1.0, d, x).
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double alpha, double lambda, int d, double x)```
Returns an approximation of the gamma distribution function with parameters α = alpha and λ = lambda.
`static double` ```cdf(double alpha, int d, double x)```
Equivalent to cdf (alpha, 1.0, d, x).
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double alpha, double lambda, double x)```
Computes the density function at x.
` double` `getAlpha()`
Return the parameter α for this object.
`static GammaDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of a gamma distribution with parameters α and λ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getLambda()`
Return the parameter λ for this object.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double alpha, double lambda)```
Computes and returns the mean E[X] = α/λ of the gamma distribution with parameters α and λ.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (α, λ) of the gamma distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double[]` `getParams()`
Return a table containing the parameters of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double alpha, double lambda)```
Computes and returns the standard deviation of the gamma distribution with parameters α and λ.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double alpha, double lambda)```
Computes and returns the variance Var[X] = α/λ2 of the gamma distribution with parameters α and λ.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double alpha, double lambda, int d, double u)```
Computes the inverse distribution function.
`static double` ```inverseF(double alpha, int d, double u)```
Same as `inverseF` (alpha, 1, d, u).
` void` ```setParams(double alpha, double lambda, int d)```

` 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

`public GammaDist(double alpha)`
Constructs a GammaDist object with parameters α = alpha and λ = 1.

```public GammaDist(double alpha,
double lambda)```
Constructs a GammaDist object with parameters α = alpha and λ = lambda.

```public GammaDist(double alpha,
double lambda,
int d)```
Constructs a GammaDist object with parameters α = alpha and λ = lambda, and approximations of roughly d decimal digits of precision when computing functions.

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 lambda,
double x)```
Computes the density function at x.

### cdf

```public static double cdf(double alpha,
double lambda,
int d,
double x)```
Returns an approximation of the gamma distribution function with parameters α = alpha and λ = lambda. The function tries to return d decimals digits of precision. For α not too large (e.g., α <= 1000), d gives a good idea of the precision attained.

### cdf

```public static double cdf(double alpha,
int d,
double x)```
Equivalent to cdf (alpha, 1.0, d, x).

### barF

```public static double barF(double alpha,
double lambda,
int d,
double x)```
Computes the complementary distribution function.

### barF

```public static double barF(double alpha,
int d,
double x)```
Same as `barF` (alpha, 1.0, d, x).

### inverseF

```public static double inverseF(double alpha,
double lambda,
int d,
double u)```
Computes the inverse distribution function.

### inverseF

```public static double inverseF(double alpha,
int d,
double u)```
Same as `inverseF` (alpha, 1, d, u).

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (α, λ) of the gamma 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, λ].

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(λ)]

### getInstanceFromMLE

```public static GammaDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of a gamma distribution with parameters α 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 alpha,
double lambda)```
Computes and returns the mean E[X] = α/λ of the gamma distribution with parameters α and λ.

Returns:
the mean of the gamma distribution E[X] = α/λ

### getVariance

```public static double getVariance(double alpha,
double lambda)```
Computes and returns the variance Var[X] = α/λ2 of the gamma distribution with parameters α and λ.

Returns:
the variance of the gamma distribution Var[X] = α/λ2

### getStandardDeviation

```public static double getStandardDeviation(double alpha,
double lambda)```
Computes and returns the standard deviation of the gamma distribution with parameters α and λ.

Returns:
the standard deviation of the gamma distribution

### getAlpha

`public double getAlpha()`
Return the parameter α for this object.

### getLambda

`public double getLambda()`
Return the parameter λ for this object.

### setParams

```public void setParams(double alpha,
double lambda,
int d)```

### getParams

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