Class PascalDistribution
 java.lang.Object

 org.hipparchus.distribution.discrete.AbstractIntegerDistribution

 org.hipparchus.distribution.discrete.PascalDistribution

 All Implemented Interfaces:
Serializable
,IntegerDistribution
public class PascalDistribution extends AbstractIntegerDistribution
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,
wherer
is the number of successes,p
is the probability of success, andX
is the total number of failures.C(n, k)
is the binomial coefficient (n
choosek
). The mean and variance ofX
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.


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

Method Summary
All Methods Instance Methods Concrete Methods Modifier and Type Method Description double
cumulativeProbability(int x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.int
getNumberOfSuccesses()
Access the number of successes for this distribution.double
getNumericalMean()
Use this method to get the numerical value of the mean of this distribution.double
getNumericalVariance()
Use this method to get the numerical value of the variance of this distribution.double
getProbabilityOfSuccess()
Access the probability of success for this distribution.int
getSupportLowerBound()
Access the lower bound of the support.int
getSupportUpperBound()
Access the upper bound of the support.boolean
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.double
logProbability(int x)
For a random variableX
whose values are distributed according to this distribution, this method returnslog(P(X = x))
, wherelog
is the natural logarithm.double
probability(int x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X = x)
.
Methods inherited from class org.hipparchus.distribution.discrete.AbstractIntegerDistribution
inverseCumulativeProbability, probability, solveInverseCumulativeProbability




Constructor Detail

PascalDistribution
public PascalDistribution(int r, double p) throws MathIllegalArgumentException
Create a Pascal distribution with the given number of successes and probability of success. Parameters:
r
 Number of successes.p
 Probability of success. Throws:
MathIllegalArgumentException
 if the number of successes is not positiveMathIllegalArgumentException
 if the probability of success is not in the range[0, 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 variableX
whose values are distributed according to this distribution, this method returnsP(X = x)
. In other words, this method represents the probability mass function (PMF) for the distribution. Parameters:
x
 the 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 variableX
whose values are distributed according to this distribution, this method returnslog(P(X = x))
, wherelog
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 ofIntegerDistribution.probability(int)
.The default implementation simply computes the logarithm of
probability(x)
. Specified by:
logProbability
in interfaceIntegerDistribution
 Overrides:
logProbability
in classAbstractIntegerDistribution
 Parameters:
x
 the 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 variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
. In other words, this method represents the (cumulative) distribution function (CDF) for this distribution. Parameters:
x
 the 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

getNumericalMean
public double getNumericalMean()
Use this method to get the numerical value of the mean of this distribution. For number of successesr
and probability of successp
, the mean isr * (1  p) / p
. Returns:
 the mean or
Double.NaN
if it is not defined

getNumericalVariance
public double getNumericalVariance()
Use this method to get the numerical value of the variance of this distribution. For number of successesr
and probability of successp
, the variance isr * (1  p) / p^2
. Returns:
 the variance (possibly
Double.POSITIVE_INFINITY
orDouble.NaN
if it is not defined)

getSupportLowerBound
public int getSupportLowerBound()
Access the lower bound of the support. This method must return the same value asinverseCumulativeProbability(0)
. In other words, this method must return
The lower bound of the support is always 0 no matter the parameters.inf {x in Z  P(X <= x) > 0}
. Returns:
 lower bound of the support (always 0)

getSupportUpperBound
public int getSupportUpperBound()
Access the upper bound of the support. This method must return the same value asinverseCumulativeProbability(1)
. In other words, this method must return
The upper bound of the support is always positive infinity no matter the parameters. Positive infinity is symbolized byinf {x in R  P(X <= x) = 1}
.Integer.MAX_VALUE
. Returns:
 upper bound of the support (always
Integer.MAX_VALUE
for positive infinity)

isSupportConnected
public boolean 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:
true

