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Meta.Numerics.Statistics.Distributions.ParetoDistribution Class Reference

Represents a Pareto or power law distribution. More...

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

Public Member Functions

 ParetoDistribution (double mu, double alpha)
 Initializes a new Pareto distribution. 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 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 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...
 
- Public Member Functions inherited from Meta.Numerics.Statistics.Distributions.Distribution
virtual double ExpectationValue (Func< double, double > f)
 Computes the expectation value of the given function. More...
 
virtual double GetRandomValue (Random rng)
 Returns a random value. More...
 
- Public Member Functions inherited from Meta.Numerics.Statistics.Distributions.UnivariateDistribution
virtual double Cumulant (int r)
 Computes a cumulant of the distribution. More...
 

Properties

double ScaleParameter [get]
 Gets the scale parameter of the Pareto distribution. More...
 
double ShapeParameter [get]
 Gets the shape parameter of the Pareto distribution. More...
 
double GiniCoefficient [get]
 Gets the Gini coefficient corresponding to the distribution. More...
 
override double Mean [get]
 
override double Median [get]
 
override double StandardDeviation [get]
 
override double Variance [get]
 
override double Skewness [get]
 
override Interval Support [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 LeftProbabilitySeries (double r)
 

Private Attributes

readonly double m
 
readonly double a
 

Detailed Description

Represents a Pareto or power law distribution.

Constructor & Destructor Documentation

Meta.Numerics.Statistics.Distributions.ParetoDistribution.ParetoDistribution ( double  mu,
double  alpha 
)
inline

Initializes a new Pareto distribution.

Parameters
muThe scale parameter, which must be positive.
alphaThe shape parameter, which must be positive.

Member Function Documentation

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.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

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

References Meta.Numerics.MoreMath.Pow().

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.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.Distribution.

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.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.

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.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.ParetoDistribution.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.

double Meta.Numerics.Statistics.Distributions.ParetoDistribution.LeftProbabilitySeries ( double  r)
inlineprivate
override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.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.ParetoDistribution.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.

Member Data Documentation

readonly double Meta.Numerics.Statistics.Distributions.ParetoDistribution.m
private
readonly double Meta.Numerics.Statistics.Distributions.ParetoDistribution.a
private

Property Documentation

double Meta.Numerics.Statistics.Distributions.ParetoDistribution.ScaleParameter
get

Gets the scale parameter of the Pareto distribution.

double Meta.Numerics.Statistics.Distributions.ParetoDistribution.ShapeParameter
get

Gets the shape parameter of the Pareto distribution.

For a given shape parameter &#x3B1;, the probability density falls off as x-(&#x3B1;+1).

double Meta.Numerics.Statistics.Distributions.ParetoDistribution.GiniCoefficient
get

Gets the Gini coefficient corresponding to the distribution.

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

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.Median
get

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.StandardDeviation
get

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.Variance
get

override double Meta.Numerics.Statistics.Distributions.ParetoDistribution.Skewness
get

override Interval Meta.Numerics.Statistics.Distributions.ParetoDistribution.Support
get


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