umontreal.iro.lecuyer.probdist
Class NormalDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.NormalDist

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

NormalDistQuick

public class NormalDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the normal distribution (e.g.,). It has mean μ and variance σ². Its density function is

f (x) = e^{-(x-μ)2/(2σ2)}/((2π)^1/2σ) for - ∞ < x < ∞,

where σ > 0. When μ = 0 and σ = 1, we have the standard normal distribution, with corresponding distribution function

F(x) = Φ(x) = ∫_-∞^xe^-t2/2 dt/(2π)^1/2 for - ∞ < x < ∞.

The non-static methods cdf, barF, and inverseF are implemented via cdf01, barF01, and inverseF01, respectively.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

NormalDist()
Constructs a NormalDist object with default parameters μ = 0 and σ = 1.

NormalDist(double mu, double sigma)
Constructs a NormalDist object with mean μ = mu and standard deviation σ = sigma.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double mu, double sigma, double x) Computes the complementary normal distribution function bar(F)(x) = 1 - Φ((x - μ)/σ), with mean μ and variance σ2.
static double	barF01(double x) Same as barF (0, 1, x).
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double mu, double sigma, double x) Computes the normal distribution function with mean μ and variance σ2.
static double	cdf01(double x) Same as cdf (0, 1, x).
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double mu, double sigma, double x) Computes the normal density function.
static double	density01(double x) Same as density (0, 1, x).
static NormalDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a normal distribution with parameters μ and σ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getMean() Returns the mean.
static double	getMean(double mu, double sigma) Computes and returns the mean E[X] = μ of the normal distribution with parameters μ and σ.
static double[]	getMLE(double[] x, int n) Estimates the parameters (μ, σ) of the normal distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double	getMu() Returns the parameter μ.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getSigma() Returns the parameter σ.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double mu, double sigma) Computes and returns the standard deviation σ of the normal distribution with parameters μ and σ.
double	getVariance() Returns the variance.
static double	getVariance(double mu, double sigma) Computes and returns the variance Var[X] = σ2 of the normal distribution with parameters μ and σ.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double mu, double sigma, double u) Computes the inverse normal distribution function with mean μ and variance σ2.
static double	inverseF01(double u) Same as inverseF (0, 1, u).
void	setParams(double mu, double sigma) Sets the parameters μ and σ of this object.
String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

NormalDist

public NormalDist()

Constructs a NormalDist object with default parameters μ = 0 and σ = 1.

NormalDist

public NormalDist(double mu,
                  double sigma)

Constructs a NormalDist object with mean μ = mu and standard deviation σ = sigma.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density01

public static double density01(double x)

Same as density (0, 1, x).

density

public static double density(double mu,
                             double sigma,
                             double x)

Computes the normal density function.

cdf01

public static double cdf01(double x)

Same as cdf (0, 1, x).

cdf

public static double cdf(double mu,
                         double sigma,
                         double x)

Computes the normal distribution function with mean μ and variance σ². Uses the Chebyshev approximation , which gives 16 decimals of precision.

barF01

public static double barF01(double x)

Same as barF (0, 1, x).

barF

public static double barF(double mu,
                          double sigma,
                          double x)

Computes the complementary normal distribution function bar(F)(x) = 1 - Φ((x - μ)/σ), with mean μ and variance σ². Uses a Chebyshev series giving 16 decimal digits of precision.

inverseF01

public static double inverseF01(double u)

Same as inverseF (0, 1, u).

inverseF

public static double inverseF(double mu,
                              double sigma,
                              double u)

Computes the inverse normal distribution function with mean μ and variance σ². Uses different rational Chebyshev approximations. Returns 16 decimal digits of precision for 2.2×10^-308 < u < 1.

getMLE

public static double[] getMLE(double[] x,
                              int n)

Estimates the parameters (μ, σ) of the normal distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [hat(μ), hat(σ)].

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

returns the parameters [hat(μ), hat(σ)]

getInstanceFromMLE

public static NormalDist getInstanceFromMLE(double[] x,
                                            int n)

Creates a new instance of a normal distribution with parameters μ and σ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

getMean

public static double getMean(double mu,
                             double sigma)

Computes and returns the mean E[X] = μ of the normal distribution with parameters μ and σ.

Returns:

the mean of the normal distribution E[X] = μ

getVariance

public static double getVariance(double mu,
                                 double sigma)

Computes and returns the variance Var[X] = σ² of the normal distribution with parameters μ and σ.

Returns:

the variance of the normal distribution Var[X] = σ²

getStandardDeviation

public static double getStandardDeviation(double mu,
                                          double sigma)

Computes and returns the standard deviation σ of the normal distribution with parameters μ and σ.

Returns:

the standard deviation of the normal distribution

getMu

public double getMu()

Returns the parameter μ.

getSigma

public double getSigma()

Returns the parameter σ.

setParams

public void setParams(double mu,
                      double sigma)

Sets the parameters μ and σ of this object.

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution. This table is put in regular order: [μ, σ].

toString

public String toString()

Overrides:

toString in class Object