public class ParetoDistribution extends AbstractRealDistribution
Parameters:
The probability distribution function of X
is given by (for x >= k
):
α * k^α / x^(α + 1)
k
is the scale parameter: this is the minimum possible value of X
,α
is the shape parameter: this is the Pareto indexModifier and Type | Field and Description |
---|---|
static double |
DEFAULT_INVERSE_ABSOLUTE_ACCURACY
Default inverse cumulative probability accuracy.
|
random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY
Constructor and Description |
---|
ParetoDistribution()
Create a Pareto distribution with a scale of
1 and a shape of 1 . |
ParetoDistribution(double scale,
double shape)
Create a Pareto distribution using the specified scale and shape.
|
ParetoDistribution(double scale,
double shape,
double inverseCumAccuracy)
Create a Pareto distribution using the specified scale, shape and
inverse cumulative distribution accuracy.
|
ParetoDistribution(RandomGenerator rng,
double scale,
double shape)
Creates a log-normal distribution.
|
ParetoDistribution(RandomGenerator rng,
double scale,
double shape,
double inverseCumAccuracy)
Creates a log-normal distribution.
|
Modifier and Type | Method and Description |
---|---|
double |
cumulativeProbability(double x)
For a random variable
X whose values are distributed according
to this distribution, this method returns P(X <= x) . |
double |
cumulativeProbability(double x0,
double x1)
Deprecated.
|
double |
density(double x)
Returns the probability density function (PDF) of this distribution
evaluated at the specified point
x . |
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 |
getScale()
Returns the scale parameter of this distribution.
|
double |
getShape()
Returns the shape parameter of this distribution.
|
protected double |
getSolverAbsoluteAccuracy()
Returns the solver absolute accuracy for inverse cumulative computation.
|
double |
getSupportLowerBound()
Access the lower bound of the support.
|
double |
getSupportUpperBound()
Access the upper bound of the support.
|
boolean |
isSupportConnected()
Use this method to get information about whether the support is connected,
i.e.
|
boolean |
isSupportLowerBoundInclusive()
Whether or not the lower bound of support is in the domain of the density
function.
|
boolean |
isSupportUpperBoundInclusive()
Whether or not the upper bound of support is in the domain of the density
function.
|
double |
logDensity(double x)
Returns the natural logarithm of the probability density function (PDF) of this distribution
evaluated at the specified point
x . |
double |
sample()
Generate a random value sampled from this distribution.
|
inverseCumulativeProbability, probability, probability, reseedRandomGenerator, sample
public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY
public ParetoDistribution()
1
and a shape of 1
.public ParetoDistribution(double scale, double shape) throws NotStrictlyPositiveException
scale
- the scale parameter of this distributionshape
- the shape parameter of this distributionNotStrictlyPositiveException
- if scale <= 0
or shape <= 0
.public ParetoDistribution(double scale, double shape, double inverseCumAccuracy) throws NotStrictlyPositiveException
scale
- the scale parameter of this distributionshape
- the shape parameter of this distributioninverseCumAccuracy
- Inverse cumulative probability accuracy.NotStrictlyPositiveException
- if scale <= 0
or shape <= 0
.public ParetoDistribution(RandomGenerator rng, double scale, double shape) throws NotStrictlyPositiveException
rng
- Random number generator.scale
- Scale parameter of this distribution.shape
- Shape parameter of this distribution.NotStrictlyPositiveException
- if scale <= 0
or shape <= 0
.public ParetoDistribution(RandomGenerator rng, double scale, double shape, double inverseCumAccuracy) throws NotStrictlyPositiveException
rng
- Random number generator.scale
- Scale parameter of this distribution.shape
- Shape parameter of this distribution.inverseCumAccuracy
- Inverse cumulative probability accuracy.NotStrictlyPositiveException
- if scale <= 0
or shape <= 0
.public double getScale()
public double getShape()
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.
For scale k
, and shape α
of this distribution, the PDF
is given by
0
if x < k
,α * k^α / x^(α + 1)
otherwise.x
- the point at which the PDF is evaluatedx
public double logDensity(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. 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
RealDistribution.density(double)
. The default implementation simply computes the logarithm of
density(x)
.
See documentation of density(double)
for computation details.logDensity
in class AbstractRealDistribution
x
- the point at which the PDF is evaluatedx
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.
For scale k
, and shape α
of this distribution, the CDF is given by
0
if x < k
,1 - (k / x)^α
otherwise.x
- the point at which the CDF is evaluatedx
@Deprecated public double cumulativeProbability(double x0, double x1) throws NumberIsTooLargeException
RealDistribution.cumulativeProbability(double,double)
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)
cumulativeProbability
in interface RealDistribution
cumulativeProbability
in class AbstractRealDistribution
x0
- the exclusive lower boundx1
- the inclusive upper boundx0
and x1
,
excluding the lower and including the upper endpointNumberIsTooLargeException
- if x0 > x1
protected double getSolverAbsoluteAccuracy()
getSolverAbsoluteAccuracy
in class AbstractRealDistribution
public double getNumericalMean()
For scale k
and shape α
, the mean is given by
∞
if α <= 1
,α * k / (α - 1)
otherwise.Double.NaN
if it is not definedpublic double getNumericalVariance()
For scale k
and shape α
, the variance is given by
∞
if 1 < α <= 2
,k^2 * α / ((α - 1)^2 * (α - 2))
otherwise.Double.POSITIVE_INFINITY
as
for certain cases in TDistribution
) or Double.NaN
if it
is not definedpublic double getSupportLowerBound()
inverseCumulativeProbability(0)
. In other words, this
method must return
inf {x in R | P(X <= x) > 0}
.
The lower bound of the support is equal to the scale parameter k
.
public double getSupportUpperBound()
inverseCumulativeProbability(1)
. In other words, this
method must return
inf {x in R | P(X <= x) = 1}
.
The upper bound of the support is always positive infinity no matter the parameters.
Double.POSITIVE_INFINITY
)public boolean isSupportLowerBoundInclusive()
getSupporLowerBound()
is finite and
density(getSupportLowerBound())
returns a non-NaN, non-infinite
value.public boolean isSupportUpperBoundInclusive()
getSupportUpperBound()
is finite and
density(getSupportUpperBound())
returns a non-NaN, non-infinite
value.public boolean isSupportConnected()
The support of this distribution is connected.
true
public double sample()
sample
in interface RealDistribution
sample
in class AbstractRealDistribution
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