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## cern.jet.stat
Class 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.

Version:
  0.91, 08-Dec-99

Method Summary
`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).

  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 `talpha[size]`.

  The function uses the studentT function to determine the return value iteratively.

  Parameters:
    `alpha` - probability
    `size` - size of data set

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