SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class StudentDist

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

`public class StudentDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the Student t-distribution with n degrees of freedom, where n is a positive integer. Its density is

f (x) = [Γ((n + 1)/2)/(Γ(n/2)(πn)1/2)][1 + x2/n]-(n+1)/2        for - ∞ < x < ∞,

where Γ(x) is the gamma function defined in `GammaDist`.

Field Summary

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

Constructor Summary
`StudentDist(int n)`
Constructs a StudentDist object with n degrees of freedom.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(int n, double x)```
Computes the complementary distribution function v = bar(F)(x) with n degrees of freedom.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(int n, double x)```
Computes the Student t-distribution function u = F(x) with n degrees of freedom.
`static double` ```cdf2(int n, int d, double x)```
Deprecated.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(int n, double x)```
Computes the density function of a Student t-distribution with n degrees of freedom.
`static StudentDist` ```getInstanceFromMLE(double[] x, int m)```
Creates a new instance of a Student t-distribution with parameter n estimated using the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
` double` `getMean()`
Returns the mean.
`static double` `getMean(int n)`
Returns the mean E[X] = 0 of the Student t-distribution with parameter n.
`static double[]` ```getMLE(double[] x, int m)```
Estimates the parameter n of the Student t-distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
` int` `getN()`
Returns the parameter n associated with this object.
` double[]` `getParams()`
Return a table containing the parameter of the current distribution.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` `getStandardDeviation(int n)`
Computes and returns the standard deviation of the Student t-distribution with parameter n.
` double` `getVariance()`
Returns the variance.
`static double` `getVariance(int n)`
Computes and returns the variance Var[X] = n/(n - 2) of the Student t-distribution with parameter n.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(int n, double u)```
Returns the inverse x = F-1(u) of Student t-distribution function with n degrees of freedom.
` void` `setN(int n)`
Sets the parameter n associated with 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

### StudentDist

`public StudentDist(int n)`
Constructs a StudentDist object with n degrees of freedom.

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(int n,
double x)```
Computes the density function of a Student t-distribution with n degrees of freedom.

### cdf

```public static double cdf(int n,
double x)```
Computes the Student t-distribution function u = F(x) with n degrees of freedom. Gives 13 decimal digits of precision for n <= 105. For n > 105, gives at least 6 decimal digits of precision everywhere, and at least 9 decimal digits of precision for all u > 10-15.

### cdf2

```@Deprecated
public static double cdf2(int n,
int d,
double x)```
Deprecated.

Same as `cdf`(n, x).

### barF

```public static double barF(int n,
double x)```
Computes the complementary distribution function v = bar(F)(x) with n degrees of freedom. Gives 13 decimal digits of precision for n <= 105. For n > 105, gives at least 6 decimal digits of precision everywhere, and at least 9 decimal digits of precision for all v > 10-15.

### inverseF

```public static double inverseF(int n,
double u)```
Returns the inverse x = F-1(u) of Student t-distribution function with n degrees of freedom. Gives 13 decimal digits of precision for n <= 105, and at least 9 decimal digits of precision for n > 105.

### getMLE

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

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

### getInstanceFromMLE

```public static StudentDist getInstanceFromMLE(double[] x,
int m)```
Creates a new instance of a Student t-distribution with parameter n estimated using the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.

Parameters:
`x` - the list of observations to use to evaluate parameters
`m` - the number of observations to use to evaluate parameters

### getMean

`public static double getMean(int n)`
Returns the mean E[X] = 0 of the Student t-distribution with parameter n.

Returns:
the mean of the Student t-distribution E[X] = 0

### getVariance

`public static double getVariance(int n)`
Computes and returns the variance Var[X] = n/(n - 2) of the Student t-distribution with parameter n.

Returns:
the variance of the Student t-distribution Var[X] = n/(n - 2)

### getStandardDeviation

`public static double getStandardDeviation(int n)`
Computes and returns the standard deviation of the Student t-distribution with parameter n.

Returns:
the standard deviation of the Student t-distribution

### getN

`public int getN()`
Returns the parameter n associated with this object.

### setN

`public void setN(int n)`
Sets the parameter n associated with this object.

### getParams

`public double[] getParams()`
Return a table containing the parameter 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.