Class ExponentialDistribution
- java.lang.Object
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- org.apache.commons.statistics.distribution.AbstractContinuousDistribution
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- org.apache.commons.statistics.distribution.ExponentialDistribution
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- All Implemented Interfaces:
ContinuousDistribution
public final class ExponentialDistribution extends AbstractContinuousDistribution
Implementation of the exponential distribution.The probability density function of \( X \) is:
\[ f(x; \mu) = \frac{1}{\mu} e^{-x / \mu} \]
for \( \mu > 0 \) the mean and \( x \in [0, \infty) \).
This implementation uses the scale parameter \( \mu \) which is the mean of the distribution. A common alternative parameterization uses the rate parameter \( \lambda \) which is the reciprocal of the mean. The distribution can be be created using \( \mu = \frac{1}{\lambda} \).
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Nested Class Summary
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Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution
ContinuousDistribution.Sampler
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Field Summary
Fields Modifier and Type Field Description private double
logMean
The logarithm of the mean, stored to reduce computing time.private double
mean
The mean of this distribution.private static double
SUPPORT_HI
Support upper bound.private static double
SUPPORT_LO
Support lower bound.
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Constructor Summary
Constructors Modifier Constructor Description private
ExponentialDistribution(double mean)
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description ContinuousDistribution.Sampler
createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.double
cumulativeProbability(double x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X <= x)
.double
density(double x)
Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
.double
getMean()
Gets the mean of this distribution.(package private) double
getMedian()
Gets the median.double
getSupportLowerBound()
Gets the lower bound of the support.double
getSupportUpperBound()
Gets the upper bound of the support.double
getVariance()
Gets the variance of this distribution.double
inverseCumulativeProbability(double p)
Computes the quantile function of this distribution.double
inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution.double
logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.static ExponentialDistribution
of(double mean)
Creates an exponential distribution.double
survivalProbability(double x)
For a random variableX
whose values are distributed according to this distribution, this method returnsP(X > x)
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Methods inherited from class org.apache.commons.statistics.distribution.AbstractContinuousDistribution
isSupportConnected, probability
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Field Detail
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SUPPORT_LO
private static final double SUPPORT_LO
Support lower bound.- See Also:
- Constant Field Values
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SUPPORT_HI
private static final double SUPPORT_HI
Support upper bound.- See Also:
- Constant Field Values
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mean
private final double mean
The mean of this distribution.
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logMean
private final double logMean
The logarithm of the mean, stored to reduce computing time.
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Method Detail
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of
public static ExponentialDistribution of(double mean)
Creates an exponential distribution.- Parameters:
mean
- Mean of this distribution. This is a scale parameter.- Returns:
- the distribution
- Throws:
java.lang.IllegalArgumentException
- ifmean <= 0
.
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density
public double density(double x)
Returns the probability density function (PDF) of this distribution evaluated at the specified pointx
. In general, the PDF is the derivative of the CDF. If the derivative does not exist atx
, then an appropriate replacement should be returned, e.g.Double.POSITIVE_INFINITY
,Double.NaN
, or the limit inferior or limit superior of the difference quotient.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the value of the probability density function at
x
.
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logDensity
public double logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified pointx
.- Parameters:
x
- Point at which the PDF is evaluated.- Returns:
- the logarithm of the value of the probability density function
at
x
.
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cumulativeProbability
public double cumulativeProbability(double 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(double 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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inverseCumulativeProbability
public double inverseCumulativeProbability(double p)
Computes the quantile function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]
The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
forp = 0
,ContinuousDistribution.getSupportUpperBound()
forp = 1
, or- the result of a search for a root between the lower and upper bound using
cumulativeProbability(x) - p
. The bounds may be bracketed for efficiency.
Returns
0
whenp == 0
andDouble.POSITIVE_INFINITY
whenp == 1
.- Specified by:
inverseCumulativeProbability
in interfaceContinuousDistribution
- Overrides:
inverseCumulativeProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Cumulative probability.- Returns:
- the smallest
p
-quantile of this distribution (largest 0-quantile forp = 0
).
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inverseSurvivalProbability
public double inverseSurvivalProbability(double p)
Computes the inverse survival probability function of this distribution. For a random variableX
distributed according to this distribution, the returned value is:\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
By default, this is defined as
inverseCumulativeProbability(1 - p)
, but the specific implementation may be more accurate.The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
forp = 1
,ContinuousDistribution.getSupportUpperBound()
forp = 0
, or- the result of a search for a root between the lower and upper bound using
survivalProbability(x) - p
. The bounds may be bracketed for efficiency.
Returns
0
whenp == 1
andDouble.POSITIVE_INFINITY
whenp == 0
.- Specified by:
inverseSurvivalProbability
in interfaceContinuousDistribution
- Overrides:
inverseSurvivalProbability
in classAbstractContinuousDistribution
- Parameters:
p
- Survival probability.- Returns:
- the smallest
(1-p)
-quantile of this distribution (largest 0-quantile forp = 1
).
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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.For mean \( \mu \), the variance is \( \mu^2 \).
- Returns:
- the variance.
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getSupportLowerBound
public double getSupportLowerBound()
Gets the lower bound of the support. It must return the same value asinverseCumulativeProbability(0)
, i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).The lower bound of the support is always 0.
- Returns:
- 0.
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getSupportUpperBound
public double getSupportUpperBound()
Gets the upper bound of the support. It must return the same value asinverseCumulativeProbability(1)
, i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).The upper bound of the support is always positive infinity.
- Returns:
- positive infinity.
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getMedian
double getMedian()
Gets the median. This is used to determine if the arguments to theAbstractContinuousDistribution.probability(double, double)
function are in the upper or lower domain.The default implementation calls
AbstractContinuousDistribution.inverseCumulativeProbability(double)
with a value of 0.5.- Overrides:
getMedian
in classAbstractContinuousDistribution
- Returns:
- the median
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createSampler
public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
Creates a sampler.- Specified by:
createSampler
in interfaceContinuousDistribution
- Overrides:
createSampler
in classAbstractContinuousDistribution
- Parameters:
rng
- Generator of uniformly distributed numbers.- Returns:
- a sampler that produces random numbers according this distribution.
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