Probability

java.lang.Object
- cern.jet.math.Constants
- - cern.jet.stat.Probability

```
public class Probability
extends Constants
```
Custom tailored numerical integration of certain probability distributions.
Implementation:
Some code taken and adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C). Most Cephes code (missing from the 2D Graph Package) directly ported.

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static double`	`beta(double a, double b, double x)` Returns the area from zero to `x` under the beta density function.
`static double`	`betaComplemented(double a, double b, double x)` Returns the area under the right hand tail (from `x` to infinity) of the beta density function.
`static double`	`binomial(int k, int n, double p)` Returns the sum of the terms `0` through `k` of the Binomial probability density.
`static double`	`binomialComplemented(int k, int n, double p)` Returns the sum of the terms `k+1` through `n` of the Binomial probability density.
`static double`	`chiSquare(double v, double x)` Returns the area under the left hand tail (from 0 to `x`) of the Chi square probability density function with `v` degrees of freedom.
`static double`	`chiSquareComplemented(double v, double x)` Returns the area under the right hand tail (from `x` to infinity) of the Chi square probability density function with `v` degrees of freedom.
`static double`	`errorFunction(double x)` Returns the error function of the normal distribution; formerly named `erf`.
`static double`	`errorFunctionComplemented(double a)` Returns the complementary Error function of the normal distribution; formerly named `erfc`.
`static double`	`gamma(double a, double b, double x)` Returns the integral from zero to `x` of the gamma probability density function.
`static double`	`gammaComplemented(double a, double b, double x)` Returns the integral from `x` to infinity of the gamma probability density function:
`static double`	`negativeBinomial(int k, int n, double p)` Returns the sum of the terms `0` through `k` of the Negative Binomial Distribution.
`static double`	`negativeBinomialComplemented(int k, int n, double p)` Returns the sum of the terms `k+1` to infinity of the Negative Binomial distribution.
`static double`	`normal(double a)` Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to `x` (assumes mean is zero, variance is one).
`static double`	`normal(double mean, double variance, double x)` Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to `x`.
`static double`	`normalInverse(double y0)` Returns the value, `x`, for which the area under the Normal (Gaussian) probability density function (integrated from minus infinity to `x`) is equal to the argument `y` (assumes mean is zero, variance is one); formerly named `ndtri`.
`static double`	`poisson(int k, double mean)` Returns the sum of the first `k` terms of the Poisson distribution.
`static double`	`poissonComplemented(int k, double mean)` Returns the sum of the terms `k+1` to `Infinity` of the Poisson distribution.
`static double`	`studentT(double k, double t)` Returns the integral from minus infinity to `t` of the Student-t distribution with `k > 0` degrees of freedom.
`static double`	`studentTInverse(double alpha, int size)` Returns the value, `t`, for which the area under the Student-t probability density function (integrated from minus infinity to `t`) is equal to `1-alpha/2`.

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

- Method Detail
  - beta
```
public static double beta(double a,
                          double b,
                          double x)
```
    Returns the area from zero to x under the beta density function.
```
                          x
            -             -
           | (a+b)       | |  a-1      b-1
 P(x)  =  ----------     |   t    (1-t)    dt
           -     -     | |
          | (a) | (b)   -
                         0
 
```
    This function is identical to the incomplete beta integral function Gamma.incompleteBeta(a, b, x). The complemented function is 1 - P(1-x) = Gamma.incompleteBeta( b, a, x );
  - betaComplemented
```
public static double betaComplemented(double a,
                                      double b,
                                      double x)
```
    Returns the area under the right hand tail (from x to infinity) of the beta density function. This function is identical to the incomplete beta integral function Gamma.incompleteBeta(b, a, x).
  - binomial
```
public static double binomial(int k,
                              int n,
                              double p)
```
    Returns the sum of the terms 0 through k of the Binomial probability density.
```
   k
   --  ( n )   j      n-j
   >   (   )  p  (1-p)
   --  ( j )
  j=0
 
```
    The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula
    y = binomial( k, n, p ) = Gamma.incompleteBeta( n-k, k+1, 1-p ).
    All arguments must be positive,
    Parameters:
    
    k - end term.
    
    n - the number of trials.
    
    p - the probability of success (must be in (0.0,1.0)).
  - binomialComplemented
```
public static double binomialComplemented(int k,
                                          int n,
                                          double p)
```
    Returns the sum of the terms k+1 through n of the Binomial probability density.
```
   n
   --  ( n )   j      n-j
   >   (   )  p  (1-p)
   --  ( j )
  j=k+1
 
```
    The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula
    y = binomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n-k, p ).
    All arguments must be positive,
    Parameters:
    
    k - end term.
    
    n - the number of trials.
    
    p - the probability of success (must be in (0.0,1.0)).
  - chiSquare
```
public static double chiSquare(double v,
                               double x)
                        throws ArithmeticException
```
    Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom.
```
                                  inf.
                                    -
                        1          | |  v/2-1  -t/2
  P( x | v )   =   -----------     |   t      e     dt
                    v/2  -       | |
                   2    | (v/2)   -
                                   x
 
```
    where x is the Chi-square variable.
    The incomplete gamma integral is used, according to the formula
    y = chiSquare( v, x ) = incompleteGamma( v/2.0, x/2.0 ).
    The arguments must both be positive.
    Parameters:
    
    v - degrees of freedom.
    
    x - integration end point.
    
    Throws:
    
    ArithmeticException
  - chiSquareComplemented
```
public static double chiSquareComplemented(double v,
                                           double x)
                                    throws ArithmeticException
```
    Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom.
```
                                  inf.
                                    -
                        1          | |  v/2-1  -t/2
  P( x | v )   =   -----------     |   t      e     dt
                    v/2  -       | |
                   2    | (v/2)   -
                                   x
 
```
    where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chiSquareComplemented( v, x ) = incompleteGammaComplement( v/2.0, x/2.0 ). The arguments must both be positive.
    Parameters:
    
    v - degrees of freedom.
    
    Throws:
    
    ArithmeticException
  - errorFunction
```
public static double errorFunction(double x)
                            throws ArithmeticException
```
    Returns the error function of the normal distribution; formerly named erf. The integral is
```
                           x 
                            -
                 2         | |          2
   erf(x)  =  --------     |    exp( - t  ) dt.
              sqrt(pi)   | |
                          -
                           0
 
```
    Implementation: For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise erf(x) = 1 - erfc(x).
    Code adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C).
    Parameters:
    
    a - the argument to the function.
    
    Throws:
    
    ArithmeticException
  - errorFunctionComplemented
```
public static double errorFunctionComplemented(double a)
                                        throws ArithmeticException
```
    Returns the complementary Error function of the normal distribution; formerly named erfc.
```
  1 - erf(x) =

                           inf. 
                             -
                  2         | |          2
   erfc(x)  =  --------     |    exp( - t  ) dt
               sqrt(pi)   | |
                           -
                            x
 
```
    Implementation: For small x, erfc(x) = 1 - erf(x); otherwise rational approximations are computed.
    Code adapted from the Java 2D Graph Package 2.4, which in turn is a port from the Cephes 2.2 Math Library (C).
    Parameters:
    
    a - the argument to the function.
    
    Throws:
    
    ArithmeticException
  - gamma
```
public static double gamma(double a,
                           double b,
                           double x)
```
    Returns the integral from zero to x of the gamma probability density function.
```
                x
        b       -
       a       | |   b-1  -at
 y =  -----    |    t    e    dt
       -     | |
      | (b)   -
               0
 
```
    The incomplete gamma integral is used, according to the relation y = Gamma.incompleteGamma( b, a*x ).
    Parameters:
    
    a - the paramater a (alpha) of the gamma distribution.
    
    b - the paramater b (beta, lambda) of the gamma distribution.
    
    x - integration end point.
  - gammaComplemented
```
public static double gammaComplemented(double a,
                                       double b,
                                       double x)
```
    Returns the integral from x to infinity of the gamma probability density function:
```
               inf.
        b       -
       a       | |   b-1  -at
 y =  -----    |    t    e    dt
       -     | |
      | (b)   -
               x
 
```
    The incomplete gamma integral is used, according to the relation
    y = Gamma.incompleteGammaComplement( b, a*x ).
    Parameters:
    
    a - the paramater a (alpha) of the gamma distribution.
    
    b - the paramater b (beta, lambda) of the gamma distribution.
    
    x - integration end point.
  - negativeBinomial
```
public static double negativeBinomial(int k,
                                      int n,
                                      double p)
```
    Returns the sum of the terms 0 through k of the Negative Binomial Distribution.
```
   k
   --  ( n+j-1 )   n      j
   >   (       )  p  (1-p)
   --  (   j   )
  j=0
 
```
    In a sequence of Bernoulli trials, this is the probability that k or fewer failures precede the n-th success.
    The terms are not computed individually; instead the incomplete beta integral is employed, according to the formula
    y = negativeBinomial( k, n, p ) = Gamma.incompleteBeta( n, k+1, p ). All arguments must be positive,
    Parameters:
    
    k - end term.
    
    n - the number of trials.
    
    p - the probability of success (must be in (0.0,1.0)).
  - negativeBinomialComplemented
```
public static double negativeBinomialComplemented(int k,
                                                  int n,
                                                  double p)
```
    Returns the sum of the terms k+1 to infinity of the Negative Binomial distribution.
```
   inf
   --  ( n+j-1 )   n      j
   >   (       )  p  (1-p)
   --  (   j   )
  j=k+1
 
```
    The terms are not computed individually; instead the incomplete beta integral is employed, according to the formula
    y = negativeBinomialComplemented( k, n, p ) = Gamma.incompleteBeta( k+1, n, 1-p ). All arguments must be positive,
    Parameters:
    
    k - end term.
    
    n - the number of trials.
    
    p - the probability of success (must be in (0.0,1.0)).
  - normal
```
public static double normal(double a)
                     throws ArithmeticException
```
    Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x (assumes mean is zero, variance is one).
```
                            x
                             -
                   1        | |          2
  normal(x)  = ---------    |    exp( - t /2 ) dt
               sqrt(2pi)  | |
                           -
                          -inf.

             =  ( 1 + erf(z) ) / 2
             =  erfc(z) / 2
 
```
    where z = x/sqrt(2). Computation is via the functions errorFunction and errorFunctionComplement.
    Throws:
    
    ArithmeticException
  - normal
```
public static double normal(double mean,
                            double variance,
                            double x)
                     throws ArithmeticException
```
    Returns the area under the Normal (Gaussian) probability density function, integrated from minus infinity to x.
```
                            x
                             -
                   1        | |                 2
  normal(x)  = ---------    |    exp( - (t-mean) / 2v ) dt
               sqrt(2pi*v)| |
                           -
                          -inf.

 
```
    where v = variance. Computation is via the functions errorFunction.
    Parameters:
    
    mean - the mean of the normal distribution.
    
    variance - the variance of the normal distribution.
    
    x - the integration limit.
    
    Throws:
    
    ArithmeticException
  - normalInverse
```
public static double normalInverse(double y0)
                            throws ArithmeticException
```
    Returns the value, x, for which the area under the Normal (Gaussian) probability density function (integrated from minus infinity to x) is equal to the argument y (assumes mean is zero, variance is one); formerly named ndtri.
    For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). There are two rational functions P/Q, one for 0 < y < exp(-32) and the other for y up to exp(-2). For larger arguments, w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
    
    Throws:
    
    ArithmeticException
  - poisson
```
public static double poisson(int k,
                             double mean)
                      throws ArithmeticException
```
    Returns the sum of the first k terms of the Poisson distribution.
```
   k         j
   --   -m  m
   >   e    --
   --       j!
  j=0
 
```
    The terms are not summed directly; instead the incomplete gamma integral is employed, according to the relation
    y = poisson( k, m ) = Gamma.incompleteGammaComplement( k+1, m ). The arguments must both be positive.
    Parameters:
    
    k - number of terms.
    
    mean - the mean of the poisson distribution.
    
    Throws:
    
    ArithmeticException
  - poissonComplemented
```
public static double poissonComplemented(int k,
                                         double mean)
                                  throws ArithmeticException
```
    Returns the sum of the terms k+1 to Infinity of the Poisson distribution.
```
  inf.       j
   --   -m  m
   >   e    --
   --       j!
  j=k+1
 
```
    The terms are not summed directly; instead the incomplete gamma integral is employed, according to the formula
    y = poissonComplemented( k, m ) = Gamma.incompleteGamma( k+1, m ). The arguments must both be positive.
    Parameters:
    
    k - start term.
    
    mean - the mean of the poisson distribution.
    
    Throws:
    
    ArithmeticException
  - studentT
```
public static double studentT(double k,
                              double t)
                       throws ArithmeticException
```
    Returns the integral from minus infinity to t of the Student-t distribution with k > 0 degrees of freedom.
```
                                      t
                                      -
                                     | |
              -                      |         2   -(k+1)/2
             | ( (k+1)/2 )           |  (     x   )
       ----------------------        |  ( 1 + --- )        dx
                     -               |  (      k  )
       sqrt( k pi ) | ( k/2 )        |
                                   | |
                                    -
                                   -inf.
 
```
    Relation to incomplete beta integral:
    1 - studentT(k,t) = 0.5 * Gamma.incompleteBeta( k/2, 1/2, z ) where z = k/(k + t**2).
    Since the function is symmetric about t=0, the area under the right tail of the density is found by calling the function with -t instead of t.
    Parameters:
    
    k - degrees of freedom.
    
    t - integration end point.
    
    Throws:
    
    ArithmeticException
  - studentTInverse
```
public static double studentTInverse(double alpha,
                                     int size)
```
    Returns the value, t, for which the area under the Student-t probability density function (integrated from minus infinity to t) is equal to 1-alpha/2. The value returned corresponds to usual Student t-distribution lookup table for t_alpha[size].
    The function uses the studentT function to determine the return value iteratively.
    
    Parameters:
    
    alpha - probability
    
    size - size of data set

Class Probability

Method Summary

Methods inherited from class java.lang.Object

Method Detail

beta

betaComplemented

binomial

binomialComplemented

chiSquare

chiSquareComplemented

errorFunction

errorFunctionComplemented

gamma

gammaComplemented

negativeBinomial

negativeBinomialComplemented

normal

normal

normalInverse

poisson

poissonComplemented

studentT

studentTInverse