Class PoissonDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.PoissonDistribution
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- All Implemented Interfaces:
DiscreteDistribution
public final class PoissonDistribution extends Object
Implementation of the Poisson distribution.The probability mass function of \( X \) is:
\[ f(k; \lambda) = \frac{\lambda^k e^{-k}}{k!} \]
for \( \lambda \in (0, \infty) \) the mean and \( k \in \{0, 1, 2, \dots\} \) the number of events.
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Nested Class Summary
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Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.DiscreteDistribution
DiscreteDistribution.Sampler
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description DiscreteDistribution.SamplercreateSampler(org.apache.commons.rng.UniformRandomProvider rng)Creates a sampler.doublecumulativeProbability(int x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X <= x).doublegetMean()Gets the mean of this distribution.intgetSupportLowerBound()Gets the lower bound of the support.intgetSupportUpperBound()Gets the upper bound of the support.doublegetVariance()Gets the variance of this distribution.intinverseCumulativeProbability(double p)Computes the quantile function of this distribution.intinverseSurvivalProbability(double p)Computes the inverse survival probability function of this distribution.doublelogProbability(int x)For a random variableXwhose values are distributed according to this distribution, this method returnslog(P(X = x)), wherelogis the natural logarithm.static PoissonDistributionof(double mean)Creates a Poisson distribution.doubleprobability(int x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X = x).doubleprobability(int x0, int x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1).doublesurvivalProbability(int x)For a random variableXwhose values are distributed according to this distribution, this method returnsP(X > x).
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Method Detail
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public static PoissonDistribution of(double mean)
Creates a Poisson distribution.- Parameters:
mean- Poisson mean.- Returns:
- the distribution
- Throws:
IllegalArgumentException- ifmean <= 0.
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probability
public double probability(int x)
For a random variableXwhose 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 variableXwhose values are distributed according to this distribution, this method returnslog(P(X = x)), wherelogis 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 variableXwhose 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 variableXwhose 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.- Returns:
- the mean.
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getVariance
public double getVariance()
Gets the variance of this distribution.The variance is equal to the mean.
- 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_VALUEshould 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_VALUEshould be substituted for positive infinity.The upper bound of the support is always positive infinity.
- Returns:
Integer.MAX_VALUE
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createSampler
public DiscreteDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.- Specified by:
createSamplerin interfaceDiscreteDistribution- Parameters:
rng- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
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probability
public double probability(int x0, int x1)For a random variableXwhose values are distributed according to this distribution, this method returnsP(x0 < X <= x1). The default implementation uses the identityP(x0 < X <= x1) = P(X <= x1) - P(X <= x0)Special cases:
- returns
0.0ifx0 == x1; - returns
probability(x1)ifx0 + 1 == x1;
- Specified by:
probabilityin interfaceDiscreteDistribution- Parameters:
x0- Lower bound (exclusive).x1- Upper bound (inclusive).- Returns:
- the probability that a random variable with this distribution
takes a value between
x0andx1, excluding the lower and including the upper endpoint.
- returns
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inverseCumulativeProbability
public int inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variableXdistributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb Z : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
If the result exceeds the range of the data type
int, thenInteger.MIN_VALUEorInteger.MAX_VALUEis returned. In this case the result ofcumulativeProbability(x)called using the returnedp-quantile may not compute the originalp.The default implementation returns:
DiscreteDistribution.getSupportLowerBound()forp = 0,DiscreteDistribution.getSupportUpperBound()forp = 1, or- the result of a binary search between the lower and upper bound using
cumulativeProbability(x). The bounds may be bracketed for efficiency.
- Specified by:
inverseCumulativeProbabilityin interfaceDiscreteDistribution- Parameters:
p- Cumulative probability.- Returns:
- the smallest
p-quantile of this distribution (largest 0-quantile forp = 0). - Throws:
IllegalArgumentException- ifp < 0orp > 1
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inverseSurvivalProbability
public int inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution. For a random variableXdistributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb Z : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb Z : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
If the result exceeds the range of the data type
int, thenInteger.MIN_VALUEorInteger.MAX_VALUEis returned. In this case the result ofsurvivalProbability(x)called using the returned(1-p)-quantile may not compute the originalp.By default, this is defined as
inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.The default implementation returns:
DiscreteDistribution.getSupportLowerBound()forp = 1,DiscreteDistribution.getSupportUpperBound()forp = 0, or- the result of a binary search between the lower and upper bound using
survivalProbability(x). The bounds may be bracketed for efficiency.
- Specified by:
inverseSurvivalProbabilityin interfaceDiscreteDistribution- Parameters:
p- Cumulative probability.- Returns:
- the smallest
(1-p)-quantile of this distribution (largest 0-quantile forp = 1). - Throws:
IllegalArgumentException- ifp < 0orp > 1
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