org.apache.commons.statistics.distribution

## Class PascalDistribution

• java.lang.Object
• org.apache.commons.statistics.distribution.PascalDistribution
• All Implemented Interfaces:
DiscreteDistribution

public class PascalDistribution
extends Object
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.

• ### Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.DiscreteDistribution

DiscreteDistribution.Sampler
• ### Constructor Summary

Constructors
Constructor and Description
PascalDistribution(int r, double p)
Create a Pascal distribution with the given number of successes and probability of success.
• ### Method Summary

All Methods
Modifier and Type Method and Description
DiscreteDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.
double cumulativeProbability(int x)
For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double getMean()
Gets the mean of this distribution.
int getNumberOfSuccesses()
Access the number of successes for this distribution.
double getProbabilityOfSuccess()
Access the probability of success for this distribution.
int getSupportLowerBound()
Gets the lower bound of the support.
int getSupportUpperBound()
Gets the upper bound of the support.
double getVariance()
Gets the variance of this distribution.
int inverseCumulativeProbability(double p)
Computes the quantile function of this distribution.
boolean isSupportConnected()
Indicates whether the support is connected, i.e.
double logProbability(int x)
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.
double probability(int x)
For a random variable X whose values are distributed according to this distribution, this method returns P(X = x).
double probability(int x0, int x1)
For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).
static int[] sample(int n, DiscreteDistribution.Sampler sampler)
Utility function for allocating an array and filling it with n samples generated by the given sampler.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### PascalDistribution

public PascalDistribution(int r,
double p)
Create a Pascal distribution with the given number of successes and probability of success.
Parameters:
r - Number of successes.
p - Probability of success.
Throws:
IllegalArgumentException - if r <= 0 or p < 0 or p > 1.
• ### Method Detail

• #### getNumberOfSuccesses

public int getNumberOfSuccesses()
Access the number of successes for this distribution.
Returns:
the number of successes.
• #### getProbabilityOfSuccess

public double getProbabilityOfSuccess()
Access the probability of success for this distribution.
Returns:
the probability of success.
• #### probability

public double probability(int x)
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 - Point at which the PMF is evaluated.
Returns:
the value of the probability mass function at x.
• #### logProbability

public double logProbability(int x)
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.
Parameters:
x - Point at which the PMF is evaluated.
Returns:
the logarithm of the value of the probability mass function at x.
• #### cumulativeProbability

public double cumulativeProbability(int x)
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 - Point at which the CDF is evaluated.
Returns:
the probability that a random variable with this distribution takes a value less than or equal to x.
• #### getMean

public double getMean()
Gets the mean of this distribution. For number of successes r and probability of success p, the mean is r * (1 - p) / p.
Returns:
the mean, or Double.NaN if it is not defined.
• #### getVariance

public double getVariance()
Gets the variance of this distribution. For number of successes r and probability of success p, the variance is r * (1 - p) / p^2.
Returns:
the variance, or Double.NaN if it is not defined.
• #### getSupportLowerBound

public int getSupportLowerBound()
Gets the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0), i.e. inf {x in Z | P(X <= x) > 0}. By convention, Integer.MIN_VALUE should be substituted for negative infinity. The lower bound of the support is always 0 no matter the parameters.
Returns:
lower bound of the support (always 0)
• #### getSupportUpperBound

public int getSupportUpperBound()
Gets the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1), i.e. inf {x in R | P(X <= x) = 1}. By convention, Integer.MAX_VALUE should be substituted for positive infinity. The upper bound of the support is always positive infinity no matter the parameters. Positive infinity is symbolized by Integer.MAX_VALUE.
Returns:
upper bound of the support (always Integer.MAX_VALUE for positive infinity)
• #### isSupportConnected

public boolean isSupportConnected()
Indicates 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:
true
• #### probability

public double probability(int x0,
int x1)
For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
Specified by:
probability in interface DiscreteDistribution
Parameters:
x0 - Lower bound (exclusive).
x1 - Upper bound (inclusive).
Returns:
the probability that a random variable with this distribution will take a value between x0 and x1, excluding the lower and including the upper endpoint.
• #### inverseCumulativeProbability

public int inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is
• inf{x in Z | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in Z | P(X<=x) > 0} for p = 0.
If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. The default implementation returns
Specified by:
inverseCumulativeProbability in interface DiscreteDistribution
Parameters:
p - Cumulative probability.
Returns:
the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
• #### sample

public static int[] sample(int n,
DiscreteDistribution.Sampler sampler)
Utility function for allocating an array and filling it with n samples generated by the given sampler.
Parameters:
n - Number of samples.
sampler - Sampler.
Returns:
an array of size n.
• #### createSampler

public DiscreteDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.
Specified by:
createSampler in interface DiscreteDistribution
Parameters:
rng - Generator of uniformly distributed numbers.
Returns:
a sampler that produces random numbers according this distribution.