NormalDist (Java Libraries for Stochastic Simulation)

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

Constructors
Constructor and Description
`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

Methods
Modifier and Type	Method and Description
`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.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`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.
`java.lang.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)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - 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 java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class NormalDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

NormalDist

NormalDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density01

density

cdf01

cdf

barF01

barF

inverseF01

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getMu

getSigma

setParams

getParams

toString