Class Probability
- All Implemented Interfaces:
Serializable
Implementation:
- See Also:
-
Field Summary
Modifier and TypeFieldDescriptionprotected static final double[]
COEFFICIENTS FOR METHOD normalInverse() *protected static final double[]
protected static final double[]
protected static final double[]
protected static final double[]
protected static final double[]
-
Constructor Summary
ModifierConstructorDescriptionprotected
Makes this class non instantiable, but still let's others inherit from it. -
Method Summary
Modifier and TypeMethodDescriptionstatic double
beta
(double a, double b) Returns the beta function of the arguments.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 x) Returns the Gamma function of the argument.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
incompleteBeta
(double aa, double bb, double xx) Returns the Incomplete Beta Function evaluated from zero to xx; formerly named ibeta.static double
incompleteGamma
(double a, double x) Returns the Incomplete Gamma function; formerly named igamma.static double
incompleteGammaComplement
(double a, double x) Returns the Complemented Incomplete Gamma function; formerly named igamc.static double
logGamma
(double x) Returns the natural logarithm of the gamma function; formerly named lgamma.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.
-
Field Details
-
P0
protected static final double[] P0COEFFICIENTS FOR METHOD normalInverse() * -
Q0
protected static final double[] Q0 -
P1
protected static final double[] P1 -
Q1
protected static final double[] Q1 -
P2
protected static final double[] P2 -
Q2
protected static final double[] Q2
-
-
Constructor Details
-
Probability
protected Probability()Makes this class non instantiable, but still let's others inherit from it.
-
-
Method Details
-
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 incompleteBeta(a, b, x). The complemented function is 1 - P(1-x) = 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 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 formulay = binomial( k, n, p ) = 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 formulay = binomialComplemented( k, n, p ) = 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
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
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
Returns the error function of the normal distribution; formerly named erf. The integral isx - 2 | | 2 erf(x) = -------- | exp( - t ) dt. sqrt(pi) | | - 0
Implementation: For 0 invalid input: '<'= |x| invalid input: '<' 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
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 = 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 relationy = 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 ) = 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 formulay = negativeBinomialComplemented( k, n, p ) = 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
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
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
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 invalid input: '<' y invalid input: '<' 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 invalid input: '<' y invalid input: '<' 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
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 relationy = poisson( k, m ) = 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
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 formulay = poissonComplemented( k, m ) = incompleteGamma( k+1, m ). The arguments must both be positive.
- Parameters:
k
- start term.mean
- the mean of the poisson distribution.- Throws:
ArithmeticException
-
studentT
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 * 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
- probabilitysize
- size of data set
-
beta
Returns the beta function of the arguments.- - | (a) | (b) beta( a, b ) = -----------. - | (a+b)
- Throws:
ArithmeticException
-
gamma
Returns the Gamma function of the argument.- Throws:
ArithmeticException
-
incompleteBeta
Returns the Incomplete Beta Function evaluated from zero to xx; formerly named ibeta.- Parameters:
aa
- the alpha parameter of the beta distribution.bb
- the beta parameter of the beta distribution.xx
- the integration end point.- Throws:
ArithmeticException
-
incompleteGamma
Returns the Incomplete Gamma function; formerly named igamma.- Parameters:
a
- the parameter of the gamma distribution.x
- the integration end point.- Throws:
ArithmeticException
-
incompleteGammaComplement
Returns the Complemented Incomplete Gamma function; formerly named igamc.- Parameters:
a
- the parameter of the gamma distribution.x
- the integration start point.- Throws:
ArithmeticException
-
logGamma
Returns the natural logarithm of the gamma function; formerly named lgamma.- Throws:
ArithmeticException
-