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,
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.
random, randomData
Constructor and Description 

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

PascalDistribution(RandomGenerator rng,
int r,
double p)
Create a Pascal distribution with the given number of successes and
probability of success.

Modifier and Type  Method and Description 

double 
cumulativeProbability(int x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(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 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) . 
cumulativeProbability, inverseCumulativeProbability, reseedRandomGenerator, sample, sample, solveInverseCumulativeProbability
public PascalDistribution(int r, double p) throws NotStrictlyPositiveException, OutOfRangeException
Note: this constructor will implicitly create an instance of
Well19937c
as random generator to be used for sampling only (see
AbstractIntegerDistribution.sample()
and AbstractIntegerDistribution.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.
r
 Number of successes.p
 Probability of success.NotStrictlyPositiveException
 if the number of successes is not positiveOutOfRangeException
 if the probability of success is not in the
range [0, 1]
.public PascalDistribution(RandomGenerator rng, int r, double p) throws NotStrictlyPositiveException, OutOfRangeException
rng
 Random number generator.r
 Number of successes.p
 Probability of success.NotStrictlyPositiveException
 if the number of successes is not positiveOutOfRangeException
 if the probability of success is not in the
range [0, 1]
.public int getNumberOfSuccesses()
public double getProbabilityOfSuccess()
public double probability(int x)
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.x
 the point at which the PMF is evaluatedx
public double logProbability(int x)
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
IntegerDistribution.probability(int)
.
The default implementation simply computes the logarithm of probability(x)
.
logProbability
in class AbstractIntegerDistribution
x
 the point at which the PMF is evaluatedx
public double cumulativeProbability(int x)
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.x
 the point at which the CDF is evaluatedx
public double getNumericalMean()
r
and probability of success p
,
the mean is r * (1  p) / p
.Double.NaN
if it is not definedpublic double getNumericalVariance()
r
and probability of success p
,
the variance is r * (1  p) / p^2
.Double.POSITIVE_INFINITY
or
Double.NaN
if it is not defined)public int getSupportLowerBound()
inverseCumulativeProbability(0)
. In other words, this
method must return
inf {x in Z  P(X <= x) > 0}
.
public int getSupportUpperBound()
inverseCumulativeProbability(1)
. In other words, this
method must return
inf {x in R  P(X <= x) = 1}
.
Integer.MAX_VALUE
.Integer.MAX_VALUE
for positive infinity)public boolean isSupportConnected()
true
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