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Meta.Numerics.Statistics.Distributions.CauchyDistribution Class Reference

Represents a Cauchy distribution. More...

+ Inheritance diagram for Meta.Numerics.Statistics.Distributions.CauchyDistribution:
+ Collaboration diagram for Meta.Numerics.Statistics.Distributions.CauchyDistribution:

Public Member Functions

 CauchyDistribution ()
 Initializes a new standard Cauchy distribution. More...
 
 CauchyDistribution (double mu, double gamma)
 Initializes a new Cauchy distribution. More...
 
override double ProbabilityDensity (double x)
 Returns the probability density at the given point.
Parameters
xThe reference point.
Returns
The probability density p(x).
The probability density function (PDF) gives the relative probability of obtaining different values. More...
 
override double Moment (int r)
 Computes a raw moment of the distribution.
Parameters
rThe order of the moment to compute.
Returns
The rth raw moment of the distribution.
See also
Moment
More...
 
override double MomentAboutMean (int r)
 Computes a central moment of the distribution.
Parameters
rThe order of the moment to compute.
Returns
The rth central moment of the distribution.
See also
Moment
More...
 
override double Cumulant (int r)
 Computes a cumulant of the distribution.
Parameters
rThe index of the cumulant to compute.
Returns
The rth cumulant of the distribution.
More...
 
override double LeftProbability (double x)
 Returns the cumulative probability to the left of (below) the given point.
Parameters
xThe reference point.
Returns
The integrated probability to obtain a result below the reference point.
The left probability function is commonly called the cumulative distribution function (CDF). More...
 
override double RightProbability (double x)
 Return the cumulative probability to the right of (above) the given point.
Parameters
xThe reference point.
Returns
The integrated probability 1-P(x1) to obtain a result above the reference point.
In survival analysis, the right probability function is commonly called the survival function, because it gives the fraction of the population remaining after the given time. More...
 
override double InverseLeftProbability (double P)
 Returns the point at which the cumulative distribution function attains a given value.
Parameters
PThe left cumulative probability P, which must lie between 0 and 1.
Returns
The value x at which LeftProbability equals P.
The inverse left probability is commonly called the quantile function. Given a quantile, it tells which variable value is the lower border of that quantile. More...
 
override double InverseRightProbability (double Q)
 Returns the point at which the right probability function attains the given value.
Parameters
QThe right cumulative probability, which must lie between 0 and 1.
Returns
The value x for which RightProbability equals Q.
More...
 
override double GetRandomValue (Random rng)
 Returns a random value.
Parameters
rngA random number generator.
Returns
A number distributed according to the distribution.
Note that the random number generator rng will be advanced by this method. The next call to its generator methods will not give the same value as it would had it not been passed to this method. More...
 
- Public Member Functions inherited from Meta.Numerics.Statistics.Distributions.Distribution
override double Moment (int r)
 Computes a raw moment of the distribution.
Parameters
rThe order of the moment to compute.
Returns
The rth raw moment of the distribution.
See also
Moment
More...
 
override double MomentAboutMean (int r)
 Computes a central moment of the distribution.
Parameters
rThe order of the moment to compute.
Returns
The rth central moment of the distribution.
See also
Moment
More...
 
virtual double ExpectationValue (Func< double, double > f)
 Computes the expectation value of the given function. More...
 

Properties

double FullWithAtHalfMaximum [get]
 Gets the full width at half maximum (FWHM) of the Cauchy distribution. More...
 
override double Mean [get]
 
override double Median [get]
 
override double Variance [get]
 
- Properties inherited from Meta.Numerics.Statistics.Distributions.Distribution
virtual double Median [get]
 Gets the median of the distribution. More...
 
virtual Interval Support [get]
 Gets the interval over which the distribution is nonvanishing. More...
 
- Properties inherited from Meta.Numerics.Statistics.Distributions.UnivariateDistribution
virtual double Mean [get]
 Gets the mean of the distribution. More...
 
virtual double Variance [get]
 Gets the variance of the distribution. More...
 
virtual double StandardDeviation [get]
 Gets the standard deviation of the distribution. More...
 
virtual double Skewness [get]
 Gets the skewness of the distribution. More...
 
virtual double ExcessKurtosis [get]
 Gets the excess kurtosis of the distribution. More...
 

Private Member Functions

double InverseProbability (double P, double Q)
 

Private Attributes

readonly double mu
 
readonly double gamma
 
readonly IDeviateGenerator cauchyRng
 

Detailed Description

Represents a Cauchy distribution.

In physical applications, the Cauchy distribution is usually called a Lorentz distribution. It models the shape of a spectral line.

The Cauchy distribution has "fat tails". In fact, it falls off at the minimum possible rate consistent with having a convergent integral. For this same reason, none of its moments (above the zeroth) are defined.

Constructor & Destructor Documentation

Meta.Numerics.Statistics.Distributions.CauchyDistribution.CauchyDistribution ( )
inline

Initializes a new standard Cauchy distribution.

Meta.Numerics.Statistics.Distributions.CauchyDistribution.CauchyDistribution ( double  mu,
double  gamma 
)
inline

Initializes a new Cauchy distribution.

Parameters
muThe centroid of the distribution.
gammaThe width parameter of the distribution.

Member Function Documentation

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.ProbabilityDensity ( double  x)
inlinevirtual

Returns the probability density at the given point.

Parameters
xThe reference point.
Returns
The probability density p(x).
The probability density function (PDF) gives the relative probability of obtaining different values.

Implements Meta.Numerics.Statistics.Distributions.Distribution.

Referenced by Test.DistributionTest.CauchyFWHM().

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.Moment ( int  r)
inlinevirtual

Computes a raw moment of the distribution.

Parameters
rThe order of the moment to compute.
Returns
The rth raw moment of the distribution.
See also
Moment

Implements Meta.Numerics.Statistics.Distributions.UnivariateDistribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.MomentAboutMean ( int  r)
inlinevirtual

Computes a central moment of the distribution.

Parameters
rThe order of the moment to compute.
Returns
The rth central moment of the distribution.
See also
Moment

Reimplemented from Meta.Numerics.Statistics.Distributions.UnivariateDistribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.Cumulant ( int  r)
inlinevirtual

Computes a cumulant of the distribution.

Parameters
rThe index of the cumulant to compute.
Returns
The rth cumulant of the distribution.

Reimplemented from Meta.Numerics.Statistics.Distributions.UnivariateDistribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.LeftProbability ( double  x)
inlinevirtual

Returns the cumulative probability to the left of (below) the given point.

Parameters
xThe reference point.
Returns
The integrated probability to obtain a result below the reference point.
The left probability function is commonly called the cumulative distribution function (CDF).

Reimplemented from Meta.Numerics.Statistics.Distributions.Distribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.RightProbability ( double  x)
inlinevirtual

Return the cumulative probability to the right of (above) the given point.

Parameters
xThe reference point.
Returns
The integrated probability 1-P(x1) to obtain a result above the reference point.
In survival analysis, the right probability function is commonly called the survival function, because it gives the fraction of the population remaining after the given time.

Reimplemented from Meta.Numerics.Statistics.Distributions.Distribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.InverseLeftProbability ( double  P)
inlinevirtual

Returns the point at which the cumulative distribution function attains a given value.

Parameters
PThe left cumulative probability P, which must lie between 0 and 1.
Returns
The value x at which LeftProbability equals P.
The inverse left probability is commonly called the quantile function. Given a quantile, it tells which variable value is the lower border of that quantile.

Reimplemented from Meta.Numerics.Statistics.Distributions.Distribution.

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.InverseRightProbability ( double  Q)
inlinevirtual

Returns the point at which the right probability function attains the given value.

Parameters
QThe right cumulative probability, which must lie between 0 and 1.
Returns
The value x for which RightProbability equals Q.

Reimplemented from Meta.Numerics.Statistics.Distributions.Distribution.

double Meta.Numerics.Statistics.Distributions.CauchyDistribution.InverseProbability ( double  P,
double  Q 
)
inlineprivate
override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.GetRandomValue ( Random  rng)
inlinevirtual

Returns a random value.

Parameters
rngA random number generator.
Returns
A number distributed according to the distribution.
Note that the random number generator rng will be advanced by this method. The next call to its generator methods will not give the same value as it would had it not been passed to this method.

Reimplemented from Meta.Numerics.Statistics.Distributions.Distribution.

Member Data Documentation

readonly double Meta.Numerics.Statistics.Distributions.CauchyDistribution.mu
private
readonly double Meta.Numerics.Statistics.Distributions.CauchyDistribution.gamma
private
readonly IDeviateGenerator Meta.Numerics.Statistics.Distributions.CauchyDistribution.cauchyRng
private

Property Documentation

double Meta.Numerics.Statistics.Distributions.CauchyDistribution.FullWithAtHalfMaximum
get

Gets the full width at half maximum (FWHM) of the Cauchy distribution.

The full-width at half maximum (FWHM) is the width of the distribution peak at half its maximum value.

Referenced by Test.DistributionTest.CauchyFWHM().

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.Mean
get

override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.Median
get
override double Meta.Numerics.Statistics.Distributions.CauchyDistribution.Variance
get


The documentation for this class was generated from the following file: