Top Description Fields Constructors Methods
org.apache.commons.math3.distribution

public Class PascalDistribution

extends AbstractIntegerDistribution
Class Inheritance
Imports
org.apache.commons.math3.exception.NotStrictlyPositiveException, .OutOfRangeException, org.apache.commons.math3.exception.util.LocalizedFormats, org.apache.commons.math3.random.RandomGenerator, .Well19937c, org.apache.commons.math3.special.Beta, org.apache.commons.math3.util.CombinatoricsUtils, .FastMath

Implementation of the Pascal distribution. The Pascal distribution is a special case of the Negative Binomial distribution where the number of successes parameter is an integer.

There are various ways to express the probability mass and distribution functions for the Pascal distribution. The present implementation represents the distribution of the number of failures before r successes occur. This is the convention adopted in e.g. MathWorld, but not in Wikipedia.

For a random variable X whose values are distributed according to this distribution, the probability mass function is given by
P(X = k) = C(k + r - 1, r - 1) * p^r * (1 - p)^k,
where r is the number of successes, p is the probability of success, and X is the total number of failures. C(n, k) is the binomial coefficient (n choose k). The mean and variance of X are
E(X) = (1 - p) * r / p, var(X) = (1 - p) * r / p^2.
Finally, the cumulative distribution function is given by
P(X <= k) = I(p, r, k + 1), where I is the regularized incomplete Beta function.

Since
1.2 (changed to concrete class in 3.0)
See Also
Negative binomial distribution (Wikipedia), Negative binomial distribution (MathWorld)

Field Summary

Modifier and TypeField and Description
private final double
log1mProbabilityOfSuccess

The value of log(1-p), where p is the probability of success, stored for faster computation.

private final double
logProbabilityOfSuccess

The value of log(p), where p is the probability of success, stored for faster computation.

private final int
numberOfSuccesses

The number of successes.

private final double
probabilityOfSuccess

The probability of success.

private static final long
serialVersionUID

Hides org.apache.commons.math3.distribution.AbstractIntegerDistribution.serialVersionUID.

Serializable version identifier.
Inherited from org.apache.commons.math3.distribution.AbstractIntegerDistribution:
randomrandomData

Constructor Summary

AccessConstructor and Description
public
PascalDistribution(int
Number of successes.
r, double
Probability of success.
p)

Create a Pascal distribution with the given number of successes and probability of success.

public
PascalDistribution(RandomGenerator
Random number generator.
rng, int
Number of successes.
r, double
Probability of success.
p)

Create a Pascal distribution with the given number of successes and probability of success.

Method Summary

Modifier and TypeMethod and Description
public double
cumulativeProbability(int
the point at which the CDF is evaluated
x)

Implements org.apache.commons.math3.distribution.IntegerDistribution.cumulativeProbability.

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).

public int

Returns:

the number of successes.
getNumberOfSuccesses()

Access the number of successes for this distribution.

public double
getNumericalMean()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getNumericalMean.

Use this method to get the numerical value of the mean of this distribution.

public double
getNumericalVariance()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getNumericalVariance.

Use this method to get the numerical value of the variance of this distribution.

public double

Returns:

the probability of success.
getProbabilityOfSuccess()

Access the probability of success for this distribution.

public int

Returns:

lower bound of the support (always 0)
getSupportLowerBound()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getSupportLowerBound.

Access the lower bound of the support.

public int

Returns:

upper bound of the support (always Integer.MAX_VALUE for positive infinity)
getSupportUpperBound()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getSupportUpperBound.

Access the upper bound of the support.

public boolean

Returns:

true
isSupportConnected()

Implements org.apache.commons.math3.distribution.IntegerDistribution.isSupportConnected.

Use this method to get information about whether the support is connected, i.e. whether all integers between the lower and upper bound of the support are included in the support.

public double
logProbability(int
the point at which the PMF is evaluated
x)

Overrides org.apache.commons.math3.distribution.AbstractIntegerDistribution.logProbability.

For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm.

public double
probability(int
the point at which the PMF is evaluated
x)

Implements org.apache.commons.math3.distribution.IntegerDistribution.probability.

For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).

Inherited from org.apache.commons.math3.distribution.AbstractIntegerDistribution:
cumulativeProbabilityinverseCumulativeProbabilityreseedRandomGeneratorsamplesamplesolveInverseCumulativeProbability

Field Detail

log1mProbabilityOfSuccessback to summary
private final double log1mProbabilityOfSuccess

The value of log(1-p), where p is the probability of success, stored for faster computation.

logProbabilityOfSuccessback to summary
private final double logProbabilityOfSuccess

The value of log(p), where p is the probability of success, stored for faster computation.

numberOfSuccessesback to summary
private final int numberOfSuccesses

The number of successes.

probabilityOfSuccessback to summary
private final double probabilityOfSuccess

The probability of success.

serialVersionUIDback to summary
private static final long serialVersionUID

Hides org.apache.commons.math3.distribution.AbstractIntegerDistribution.serialVersionUID.

Serializable version identifier.

Constructor Detail

PascalDistributionback to summary
public PascalDistribution(int r, double p) throws NotStrictlyPositiveException, OutOfRangeException

Create a Pascal distribution with the given number of successes and probability of success.

Note

this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead.

Parameters
r:int

Number of successes.

p:double

Probability of success.

Exceptions
NotStrictlyPositiveException:
if the number of successes is not positive
OutOfRangeException:
if the probability of success is not in the range [0, 1].
PascalDistributionback to summary
public PascalDistribution(RandomGenerator rng, int r, double p) throws NotStrictlyPositiveException, OutOfRangeException

Create a Pascal distribution with the given number of successes and probability of success.

Parameters
rng:RandomGenerator

Random number generator.

r:int

Number of successes.

p:double

Probability of success.

Exceptions
NotStrictlyPositiveException:
if the number of successes is not positive
OutOfRangeException:
if the probability of success is not in the range [0, 1].
Since
3.1

Method Detail

cumulativeProbabilityback to summary
public double cumulativeProbability(int x)

Implements org.apache.commons.math3.distribution.IntegerDistribution.cumulativeProbability.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.cumulativeProbability.

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

Parameters
x:int

the point at which the CDF is evaluated

Returns:double

the probability that a random variable with this distribution takes a value less than or equal to x

getNumberOfSuccessesback to summary
public int getNumberOfSuccesses()

Access the number of successes for this distribution.

Returns:int

the number of successes.

getNumericalMeanback to summary
public double getNumericalMean()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getNumericalMean.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.getNumericalMean.

Use this method to get the numerical value of the mean of this distribution. For number of successes r and probability of success p, the mean is r * (1 - p) / p.

Returns:double

the mean or Double.NaN if it is not defined

getNumericalVarianceback to summary
public double getNumericalVariance()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getNumericalVariance.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.getNumericalVariance.

Use this method to get the numerical value of the variance of this distribution. For number of successes r and probability of success p, the variance is r * (1 - p) / p^2.

Returns:double

the variance (possibly Double.POSITIVE_INFINITY or Double.NaN if it is not defined)

getProbabilityOfSuccessback to summary
public double getProbabilityOfSuccess()

Access the probability of success for this distribution.

Returns:double

the probability of success.

getSupportLowerBoundback to summary
public int getSupportLowerBound()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getSupportLowerBound.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.getSupportLowerBound.

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in Z | P(X <= x) > 0}.

The lower bound of the support is always 0 no matter the parameters.
Returns:int

lower bound of the support (always 0)

getSupportUpperBoundback to summary
public int getSupportUpperBound()

Implements org.apache.commons.math3.distribution.IntegerDistribution.getSupportUpperBound.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.getSupportUpperBound.

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

The upper bound of the support is always positive infinity no matter the parameters. Positive infinity is symbolized by Integer.MAX_VALUE.
Returns:int

upper bound of the support (always Integer.MAX_VALUE for positive infinity)

isSupportConnectedback to summary
public boolean isSupportConnected()

Implements org.apache.commons.math3.distribution.IntegerDistribution.isSupportConnected.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.isSupportConnected.

Use this method to get information about whether the support is connected, i.e. whether all integers between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:boolean

true

logProbabilityback to summary
public double logProbability(int x)

Overrides org.apache.commons.math3.distribution.AbstractIntegerDistribution.logProbability.

Doc from org.apache.commons.math3.distribution.AbstractIntegerDistribution.logProbability.

For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm. In other words, this method represents the logarithm of the probability mass function (PMF) for the distribution. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of probability(int).

The default implementation simply computes the logarithm of probability(x).

Parameters
x:int

the point at which the PMF is evaluated

Returns:double

the logarithm of the value of the probability mass function at x

Annotations
@Override
probabilityback to summary
public double probability(int x)

Implements org.apache.commons.math3.distribution.IntegerDistribution.probability.

Doc from org.apache.commons.math3.distribution.IntegerDistribution.probability.

For a random variable X whose values are distributed according to this distribution, this method returns P(X = x). In other words, this method represents the probability mass function (PMF) for the distribution.

Parameters
x:int

the point at which the PMF is evaluated

Returns:double

the value of the probability mass function at x