public final class LevyDistribution extends Object
The probability density function of \( X \) is:
\[ f(x; \mu, c) = \sqrt{\frac{c}{2\pi}}~~\frac{e^{ -\frac{c}{2(x-\mu)}}} {(x-\mu)^{3/2}} \]
for \( \mu \) the location, \( c > 0 \) the scale, and \( x \in [\mu, \infty) \).
ContinuousDistribution.Sampler
Modifier and Type | Method and Description |
---|---|
ContinuousDistribution.Sampler |
createSampler(UniformRandomProvider rng)
Creates a sampler.
|
double |
cumulativeProbability(double x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(X <= x) . |
double |
density(double x)
Returns the probability density function (PDF) of this distribution
evaluated at the specified point
x . |
double |
getLocation()
Gets the location parameter of this distribution.
|
double |
getMean()
Gets the mean of this distribution.
|
double |
getScale()
Gets the scale parameter of this distribution.
|
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 point
x . |
static LevyDistribution |
of(double mu,
double c)
Creates a Levy distribution.
|
double |
probability(double x0,
double x1)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(x0 < X <= x1) . |
double |
survivalProbability(double x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(X > x) . |
public static LevyDistribution of(double mu, double c)
mu
- Location parameter.c
- Scale parameter.IllegalArgumentException
- if c <= 0
.public double getLocation()
public double getScale()
public double density(double x)
x
.
In general, the PDF is the derivative of the CDF
.
If the derivative does not exist at x
, 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.
If x
is less than the location parameter then 0
is
returned, as in these cases the distribution is not defined.
x
- Point at which the PDF is evaluated.x
.public double logDensity(double x)
x
.x
- Point at which the PDF is evaluated.x
.public double cumulativeProbability(double 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
- Point at which the CDF is evaluated.x
.public double survivalProbability(double x)
X
whose values are distributed according
to this distribution, this method returns P(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.
x
- Point at which the survival function is evaluated.x
.public double inverseCumulativeProbability(double p)
X
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()
for p = 0
,ContinuousDistribution.getSupportUpperBound()
for p = 1
, orcumulativeProbability(x) - p
.
The bounds may be bracketed for efficiency.inverseCumulativeProbability
in interface ContinuousDistribution
p
- Cumulative probability.p
-quantile of this distribution
(largest 0-quantile for p = 0
).public double inverseSurvivalProbability(double p)
X
distributed according to this distribution, the
returned value is:
\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \ge x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \ge x) \lt 1 \} & \text{for } p = 1 \end{cases} \]
By default, this is defined as inverseCumulativeProbability(1 - p)
, but
the specific implementation may be more accurate.
The default implementation returns:
ContinuousDistribution.getSupportLowerBound()
for p = 1
,ContinuousDistribution.getSupportUpperBound()
for p = 0
, orsurvivalProbability(x) - p
.
The bounds may be bracketed for efficiency.inverseSurvivalProbability
in interface ContinuousDistribution
p
- Survival probability.(1-p)
-quantile of this distribution
(largest 0-quantile for p = 1
).public double getMean()
The mean is equal to positive infinity.
positive infinity
.public double getVariance()
The variance is equal to positive infinity.
positive infinity
.public double getSupportLowerBound()
inverseCumulativeProbability(0)
, i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).
The lower bound of the support is the location
.
public double getSupportUpperBound()
inverseCumulativeProbability(1)
, i.e.
\( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).
The upper bound of the support is always positive infinity.
positive infinity
.public ContinuousDistribution.Sampler createSampler(UniformRandomProvider rng)
createSampler
in interface ContinuousDistribution
rng
- Generator of uniformly distributed numbers.public double probability(double x0, double x1)
X
whose values are distributed according
to this distribution, this method returns P(x0 < X <= x1)
.
The default implementation uses the identity
P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
probability
in interface ContinuousDistribution
x0
- Lower bound (exclusive).x1
- Upper bound (inclusive).x0
and x1
, excluding the lower
and including the upper endpoint.Copyright © 2018–2022 The Apache Software Foundation. All rights reserved.