SSJ
V. 1.2.5.

umontreal.iro.lecuyer.probdist
Class NormalDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by 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 parameters μ = mu and σ = sigma.
 
Method Summary
 double barF(double x)
          Returns bar(F)(x) = 1 - F(x).
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 σ.
static double barF01(double x)
          Same as barF (0.0, 1.0, x).
 double cdf(double x)
          Computes and 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 (0.0, 1.0, x).
 double density(double x)
          Returns f (x), the density of X evaluated at x.
static double density(double mu, double sigma, double x)
          Computes the normal density function.
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 in table x[i], i = 0, 1,…, n - 1.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameters [hat(μ), hat(σ)] of the normal distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.
 double getMean()
          Returns the mean of the distribution function.
static double getMean(double mu, double sigma)
          Computes and returns the mean E[X] = μ of the normal distribution with parameters μ and σ.
 double getMu()
          Returns the parameter μ.
 double getSigma()
          Returns the parameter σ.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
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 of the distribution function.
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)
          Computes and returns the inverse distribution function F-1(u), defined in.
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 (0.0, 1.0, u).
 void setParams(double mu, double sigma)
          Sets the parameters μ and σ of this object.
 
Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
inverseBisection, inverseBrent
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, 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 parameters μ = mu and σ = sigma.

Method Detail

density

public double density(double x)
Description copied from class: ContinuousDistribution
Returns f (x), the density of X 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
Computes and 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 interface: Distribution
Returns 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 interface: Distribution
Computes and returns the inverse distribution function F-1(u), defined in.

Specified by:
inverseF in interface Distribution
Overrides:
inverseF in class ContinuousDistribution
Parameters:
u - value in the interval (0, 1) for which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

getMean

public double getMean()
Description copied from interface: Distribution
Returns the mean of the distribution function.


getVariance

public double getVariance()
Description copied from interface: Distribution
Returns the variance of the distribution function.


getStandardDeviation

public double getStandardDeviation()
Description copied from interface: Distribution
Returns the standard deviation of the distribution function.


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 (0.0, 1.0, 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 (0.0, 1.0, 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 σ. Uses a Chebyshev series giving 16 decimal digits of precision.


inverseF01

public static double inverseF01(double u)
Same as inverseF (0.0, 1.0, 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 rational Chebyshev approximations giving at least 16 decimal digits of precision.


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 in table 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

getMaximumLikelihoodEstimate

public static double[] getMaximumLikelihoodEstimate(double[] x,
                                                    int n)
Estimates and returns the parameters [hat(μ), hat(σ)] of the normal distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.

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)]

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.


SSJ
V. 1.2.5.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.