Class PascalDistribution
- All Implemented Interfaces:
DiscreteDistribution
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.
The probability mass function of \( X \) is:
\[ f(k; r, p) = \binom{k+r-1}{r-1} p^r \, (1-p)^k \]
for \( r \in \{1, 2, \dots\} \) the number of successes, \( p \in (0, 1] \) the probability of success, \( k \in \{0, 1, 2, \dots\} \) the total number of failures, and
\[ \binom{k+r-1}{r-1} = \frac{(k+r-1)!}{(r-1)! \, k!} \]
is the binomial coefficient.
The cumulative distribution function of \( X \) is:
\[ P(X \leq k) = I(p, r, k + 1) \]
where \( I \) is the regularized incomplete beta function.
- See Also:
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Nested Class Summary
Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.DiscreteDistribution
DiscreteDistribution.Sampler
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Field Summary
FieldsModifier and TypeFieldDescriptionprivate final double
The value oflog(1-p)
, wherep
is the probability of success, stored for faster computation.private final double
The value oflog(p) * n
, wherep
is the probability of success andn
is the number of successes, stored for faster computation.private final int
The number of successes.private final double
The probability of success.private final double
The value ofp^n
, wherep
is the probability of success andn
is the number of successes, stored for faster computation. -
Constructor Summary
Constructors -
Method Summary
Modifier and TypeMethodDescriptiondouble
cumulativeProbability
(int x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
getMean()
Gets the mean of this distribution.int
Gets the number of successes parameter of this distribution.double
Gets the probability of success parameter of this distribution.int
Gets the lower bound of the support.int
Gets the upper bound of the support.double
Gets the variance of this distribution.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.static PascalDistribution
of
(int r, double p) Create a Pascal distribution.double
probability
(int x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X = x)
.double
survivalProbability
(int x) For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
.Methods inherited from class org.apache.commons.statistics.distribution.AbstractDiscreteDistribution
createSampler, getMedian, inverseCumulativeProbability, inverseSurvivalProbability, probability
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Field Details
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numberOfSuccesses
private final int numberOfSuccessesThe number of successes. -
probabilityOfSuccess
private final double probabilityOfSuccessThe probability of success. -
logProbabilityOfSuccessByNumOfSuccesses
private final double logProbabilityOfSuccessByNumOfSuccessesThe value oflog(p) * n
, wherep
is the probability of success andn
is the number of successes, stored for faster computation. -
log1mProbabilityOfSuccess
private final double log1mProbabilityOfSuccessThe value oflog(1-p)
, wherep
is the probability of success, stored for faster computation. -
probabilityOfSuccessPowNumOfSuccesses
private final double probabilityOfSuccessPowNumOfSuccessesThe value ofp^n
, wherep
is the probability of success andn
is the number of successes, stored for faster computation.
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Constructor Details
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PascalDistribution
private PascalDistribution(int r, double p) - Parameters:
r
- Number of successes.p
- Probability of success.
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Method Details
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of
Create a Pascal distribution.- Parameters:
r
- Number of successes.p
- Probability of success.- Returns:
- the distribution
- Throws:
IllegalArgumentException
- ifr <= 0
orp <= 0
orp > 1
.
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getNumberOfSuccesses
public int getNumberOfSuccesses()Gets the number of successes parameter of this distribution.- Returns:
- the number of successes.
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getProbabilityOfSuccess
public double getProbabilityOfSuccess()Gets the probability of success parameter of this distribution.- Returns:
- the probability of success.
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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
- Point at which the PMF is evaluated.- Returns:
- the value of the probability mass function at
x
.
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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.- Parameters:
x
- Point at which the PMF is evaluated.- Returns:
- the logarithm of the value of the probability mass function at
x
.
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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
- 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
.
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survivalProbability
public double survivalProbability(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 complementary cumulative distribution function.By default, this is defined as
1 - cumulativeProbability(x)
, but the specific implementation may be more accurate.- Parameters:
x
- Point at which the survival function is evaluated.- Returns:
- the probability that a random variable with this
distribution takes a value greater than
x
.
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getMean
public double getMean()Gets the mean of this distribution.For number of successes \( r \) and probability of success \( p \), the mean is:
\[ \frac{r (1 - p)}{p} \]
- Returns:
- the mean.
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getVariance
public double getVariance()Gets the variance of this distribution.For number of successes \( r \) and probability of success \( p \), the variance is:
\[ \frac{r (1 - p)}{p^2} \]
- Returns:
- the variance.
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getSupportLowerBound
public int getSupportLowerBound()Gets the lower bound of the support. This method must return the same value asinverseCumulativeProbability(0)
, i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} \). By convention,Integer.MIN_VALUE
should be substituted for negative infinity.The lower bound of the support is always 0.
- Returns:
- 0.
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getSupportUpperBound
public int getSupportUpperBound()Gets the upper bound of the support. This method must return the same value asinverseCumulativeProbability(1)
, i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) = 1 \} \). By convention,Integer.MAX_VALUE
should be substituted for positive infinity.The upper bound of the support is positive infinity except for the probability parameter
p = 1.0
.- Returns:
Integer.MAX_VALUE
or 0.
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