Probability

java.lang.Object
- org.mathIT.statistics.Probability

```
public class Probability
extends Object
```
This class provides probabilistic functions such as probability distributions or quantiles. They are implemented as static methods.

Version:

1.2

Author:

Andreas de Vries

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static double[]`	`auxzbr(double x0, double x1, String function, Class[] argTypes, double par, int npar1, int npar2)` yields two numbers which enclose a zero of the inputted function.
`static double`	`auxzfn(double x0, double x1, String function, Class[] argTypes, double par, int npar1, int npar2, double epsiln)` Determines numerically a zero of the inputted function, enclosed by x0 and x1.
`static double`	`normalDensity(double x, double x0, double sigma)` Returns the normal density with the specified parameters.
`static double`	`normalDistribution(double x, double x0, double sigma)` Returns normal distribution.
`static double`	`normalQuantile(double p, double x0, double sigma)` Computes the quantile of the normal distribution.
`static byte`	`signProd(double x, double y)` returns sign(xy).
`static double`	`standardNormalDensity(double x)` Standard normal density function.
`static double`	`standardNormalDistribution(double x)` standard normal distribution.
`static double`	`standardNormalQuantile(double p)` Standard normal quantile.
`static double`	`studentDensity(double x, int n)` Student t-density function for n degrees of freedom.
`static double`	`studentDistribution(double x, int n)` Student t-distribution function of n degrees of freedom.
`static double`	`studentQuantile(double p, int n)` Computes the quantile of the Student t-distribution.
`static double`	`szstnr(double x, double p)` Returns p minus the cumulative standardized normal of x.
`static double`	`szstud(double x, double p, int n)` returns p minus the cumulative Student distribution of (x, n).

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Method Detail
  - normalDensity
```
public static double normalDensity(double x,
                                   double x0,
                                   double sigma)
```
    Returns the normal density with the specified parameters.
    
    Parameters:
    
    x - the current value
    
    x0 - the average value
    
    sigma - the standard deviation
    
    Returns:
    
    the standard normal density of (x - x0) / sigma
  - normalDistribution
```
public static double normalDistribution(double x,
                                        double x0,
                                        double sigma)
```
    Returns normal distribution.
    
    Parameters:
    
    x - the current value
    
    x0 - the average value
    
    sigma - the standard deviation
    
    Returns:
    
    the standard normal density of (x - x0) / sigma
  - normalQuantile
```
public static double normalQuantile(double p,
                                    double x0,
                                    double sigma)
```
    Computes the quantile of the normal distribution.
    
    Parameters:
    
    p - probability
    
    x0 - width
    
    sigma - standard deviation
    
    Returns:
    
    quantile x_p
  - standardNormalDensity
```
public static double standardNormalDensity(double x)
```
    Standard normal density function.
    
    Parameters:
    
    x - value of random variable
    
    Returns:
    
    standard normal density value
  - standardNormalDistribution
```
public static double standardNormalDistribution(double x)
```
    standard normal distribution.
    
    Parameters:
    
    x - value of random variable
    
    Returns:
    
    standard normal distribution value
  - standardNormalQuantile
```
public static double standardNormalQuantile(double p)
```
    Standard normal quantile. See http://mathworld.wolfram.com/QuantileFunction.html
    
    Parameters:
    
    p - the probability of the quantile
    
    Returns:
    
    the quantile of p
  - studentDensity
```
public static double studentDensity(double x,
                                    int n)
```
    Student t-density function for n degrees of freedom. It is defined by
```
 
                 ( 1 + t^2/n )^{-(n+1)/2}
         f(x) = --------------------------.
                   B(.5, .5*n) sqrt{n}

  
```
    where B denotes the beta function.
    Parameters:
    
    x - independent variable
    
    n - degrees of freedom
    
    Returns:
    
    value of the Student density
    
    See Also:
    
    Numbers.beta(double,double)
  - studentDistribution
```
public static double studentDistribution(double x,
                                         int n)
```
    Student t-distribution function of n degrees of freedom.
    
    Parameters:
    
    x - a real number
    
    n - degrees of freedom of the Student t-distribution
    
    Returns:
    
    the value of the Student t-distribution function with n degrees of freedom at x
    
    See Also:
    
    studentDensity(double,int)
  - studentQuantile
```
public static double studentQuantile(double p,
                                     int n)
```
    Computes the quantile of the Student t-distribution.
    
    Parameters:
    
    p - the probability
    
    n - degrees of freedom of the Student t-distribution
    
    Returns:
    
    the quantile referring to p
    
    See Also:
    
    studentDensity(double,int)
  - szstnr
```
public static double szstnr(double x,
                            double p)
```
    Returns p minus the cumulative standardized normal of x. This is an auxiliary function which is invoked via callback ("reflection") and must therefore be declared "public".
    
    Parameters:
    
    x - a random variable value
    
    p - a probability
    
    Returns:
    
    p minus the cumulative standardized normal of x
  - szstud
```
public static double szstud(double x,
                            double p,
                            int n)
```
    returns p minus the cumulative Student distribution of (x, n). This is an auxiliary function which is invoked via callback ("reflection") and must therefore be declared "public".
    
    Parameters:
    
    x - a random variable value
    
    p - a probability
    
    n - degrees of freedom
    
    Returns:
    
    p minus the cumulative student distriibution of (x, n)
  - auxzbr
```
public static double[] auxzbr(double x0,
                              double x1,
                              String function,
                              Class[] argTypes,
                              double par,
                              int npar1,
                              int npar2)
                       throws ClassNotFoundException,
                              NoSuchMethodException,
                              IllegalAccessException,
                              InvocationTargetException
```
    yields two numbers which enclose a zero of the inputted function. The function has to be either implemented in the actual class, or referred to with its full path name (package + class). It must be declared public!
    
    Parameters:
    
    x0 - first x-value in whose neighborhoood the zero is supposed
    
    x1 - second x-value in whose neighborhoood the zero is supposed
    
    function - name of function to be considered (must be declared public!)
    
    argTypes - types of the arguments
    
    par - parameter of the function
    
    npar1 - parameter of the function
    
    npar2 - parameter of the function
    
    Returns:
    
    an array x[] consisting of two enclosing x-values {x0, x1}
    
    Throws:
    
    ClassNotFoundException - if the arguments are not correct
    
    NoSuchMethodException - if the arguments are not correct
    
    IllegalAccessException - if the arguments are not correct
    
    InvocationTargetException - if the arguments are not correct
  - auxzfn
```
public static double auxzfn(double x0,
                            double x1,
                            String function,
                            Class[] argTypes,
                            double par,
                            int npar1,
                            int npar2,
                            double epsiln)
                     throws ClassNotFoundException,
                            NoSuchMethodException,
                            IllegalAccessException,
                            InvocationTargetException
```
    Determines numerically a zero of the inputted function, enclosed by x0 and x1.
    
    Parameters:
    
    x0 - first x-value in whose neighborhoood the zero is supposed
    
    x1 - second x-value in whose neighborhoood the zero is supposed
    
    function - name of function to be considered; it has to be either implemented in the actual class, or referred to with its full path name
    
    argTypes - types of the arguments
    
    par - parameter of the function
    
    npar1 - parameter of the function
    
    npar2 - parameter of the function
    
    epsiln - accuracy of approximation
    
    Returns:
    
    an array x[] consisting of two enclosing x-values {x0, x1}
    
    Throws:
    
    ClassNotFoundException - if the arguments are not correct
    
    NoSuchMethodException - if the arguments are not correct
    
    IllegalAccessException - if the arguments are not correct
    
    InvocationTargetException - if the arguments are not correct
  - signProd
```
public static byte signProd(double x,
                            double y)
```
    returns sign(xy). I.e., 0 if x = y = 0, 1 if sign(x) == sign(y), and -1 if sign(x) != sign(y).
    
    Parameters:
    
    x - a real number
    
    y - a real number
    
    Returns:
    
    sign(xy)

Class Probability

Method Summary

Methods inherited from class java.lang.Object

Method Detail

normalDensity

normalDistribution

normalQuantile

standardNormalDensity

standardNormalDistribution

standardNormalQuantile

studentDensity

studentDistribution

studentQuantile

szstnr

szstud

auxzbr

auxzfn

signProd