java.lang.Object
- jsl.utilities.distributions.Distribution
- - jsl.utilities.distributions.Normal

All Implemented Interfaces:

ControllableIfc, CDFIfc, ContinuousDistributionIfc, DistributionFunctionIfc, DistributionIfc, DomainIfc, FirstOrderLossFunctionIfc, InverseCDFIfc, LossFunctionDistributionIfc, MeanIfc, PDFIfc, SecondOrderLossFunctionIfc, VarianceIfc, GetNameIfc, IdentityIfc, NewInstanceIfc, ParametersIfc, GetRVariableIfc
```
public class Normal
extends Distribution
implements ContinuousDistributionIfc, LossFunctionDistributionIfc, InverseCDFIfc, GetRVariableIfc
```
Models normally distributed random variables

Nested Class Summary
- Nested classes/interfaces inherited from class jsl.utilities.distributions.Distribution
  Distribution.RandomControls

Field Summary
- Fields inherited from class jsl.utilities.distributions.Distribution
  myId, myName

Constructor Summary

Constructors
Constructor	Description
`Normal()`	Constructs a normal distribution with mean 0.0 and variance 1.0
`Normal(double[] parameters)`	Constructs a normal distribution with mean = parameters[0] and variance = parameters[1]
`Normal(double mean, double variance)`	Constructs a normal distribution with mean and variance.
`Normal(double mean, double variance, java.lang.String name)`	Constructs a normal distribution with mean and variance.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method	Description
`double`	`cdf(double x)`	Returns the F(x) = Pr{X <= x} where F represents the cumulative distribution function
`double`	`complementaryCDF(double x)`	Computes the complementary cumulative probability distribution function for given value of x
`double`	`firstOrderLossFunction(double x)`	Computes the first order loss function for the function for given value of x, G1(x) = E[max(X-x,0)]
`Interval`	`getDomain()`
`double`	`getKurtosis()`	Gets the kurtosis of the distribution
`double`	`getMean()`	Returns the mean or expected value of a distribution
`double[]`	`getParameters()`	Gets the parameters for the distribution
`RVariableIfc`	`getRandomVariable(RNStreamIfc rng)`
`double`	`getSkewness()`	Gets the skewness of the distribution
`double`	`getVariance()`	Returns the variance of the distribution if defined
`double`	`invCDF(double p)`	Provides the inverse cumulative distribution function for the distribution While closed form solutions for the inverse cdf may not exist, numerical search methods can be used to solve F(X) = U.
`Normal`	`newInstance()`	Returns a new instance
`double`	`pdf(double x)`	Returns the f(x) where f represents the probability density function for the distribution.
`double`	`secondOrderLossFunction(double x)`	Computes the 2nd order loss function for the distribution function for given value of x, G2(x) = (1/2)E[max(X-x,0)*max(X-x-1,0)]
`void`	`setMean(double mean)`	Sets the mean of this normal distribution
`void`	`setParameters(double[] parameters)`	Sets the parameters for the distribution mean = parameters[0] and variance = parameters[1]
`void`	`setVariance(double variance)`	Sets the variance of this normal distribution
`static double`	`stdNormalCDF(double z)`	Computes the cumulative distribution function for a standard normal distribution using Taylor approximation.
`static double`	`stdNormalCDFAbramovitzAndStegun(double z)`	Computes the cumulative distribution function for a standard normal distribution from Abramovitz and Stegun, see also Didier H.
`static double`	`stdNormalComplementaryCDF(double z)`	Computes the complementary cumulative probability for the standard normal distribution function for given value of z
`static double`	`stdNormalFirstOrderLossFunction(double z)`	Computes the first order loss function for the standard normal distribution function for given value of x, G1(z) = E[max(Z-z,0)]
`static double`	`stdNormalInvCDF(double p)`	Computes the inverse cumulative distribution function for a standard normal distribution see, W.
`static double`	`stdNormalPDF(double z)`	Computes the pdf function for a standard normal distribution from Abramovitz and Stegun, see also Didier H.
`static double`	`stdNormalSecondOrderLossFunction(double z)`	Computes the 2nd order loss function for the standard normal distribution function for given value of z, G2(z) = (1/2)E[max(Z-z,0)*max(Z-z-1,0)]

Methods inherited from class jsl.utilities.distributions.Distribution
getControls, getId, getName, getStandardDeviation, inverseContinuousCDFViaBisection, inverseContinuousCDFViaBisection, inverseDiscreteCDFViaSearchUp, setControls, setId, setName, toString

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface jsl.utilities.distributions.CDFIfc
cdf

Methods inherited from interface jsl.utilities.random.rvariable.GetRVariableIfc
getRandomVariable, getRandomVariable

Methods inherited from interface jsl.utilities.distributions.VarianceIfc
getStandardDeviation

- Constructor Detail
  - Normal
```
public Normal()
```
    Constructs a normal distribution with mean 0.0 and variance 1.0
  - Normal
```
public Normal(double[] parameters)
```
    Constructs a normal distribution with mean = parameters[0] and variance = parameters[1]
    
    Parameters:
    
    parameters - An array with the mean and variance
  - Normal
```
public Normal(double mean,
              double variance)
```
    Constructs a normal distribution with mean and variance.
    
    Parameters:
    
    mean - of the distribution
    
    variance - must be > 0
  - Normal
```
public Normal(double mean,
              double variance,
              java.lang.String name)
```
    Constructs a normal distribution with mean and variance.
    
    Parameters:
    
    mean - of the distribution
    
    variance - must be > 0
    
    name - an optional name/label
- Method Detail
  - newInstance
```
public final Normal newInstance()
```
    Description copied from interface: NewInstanceIfc
    
    Returns a new instance
    
    Specified by:
    
    newInstance in interface NewInstanceIfc
    
    Specified by:
    
    newInstance in class Distribution
    
    Returns:
    
    the new instance
  - getDomain
```
public final Interval getDomain()
```
    Specified by:
    
    getDomain in interface DomainIfc
  - setMean
```
public final void setMean(double mean)
```
    Sets the mean of this normal distribution
    
    Parameters:
    
    mean - of the distribution
  - getMean
```
public final double getMean()
```
    Description copied from interface: MeanIfc
    
    Returns the mean or expected value of a distribution
    
    Specified by:
    
    getMean in interface MeanIfc
    
    Returns:
    
    double the mean or expected value for the distribution
  - setVariance
```
public final void setVariance(double variance)
```
    Sets the variance of this normal distribution
    
    Parameters:
    
    variance - of the distribution, must be > 0
  - getVariance
```
public final double getVariance()
```
    Description copied from interface: VarianceIfc
    
    Returns the variance of the distribution if defined
    
    Specified by:
    
    getVariance in interface VarianceIfc
    
    Returns:
    
    double the variance of the random variable
  - stdNormalCDFAbramovitzAndStegun
```
public static double stdNormalCDFAbramovitzAndStegun(double z)
```
    Computes the cumulative distribution function for a standard normal distribution from Abramovitz and Stegun, see also Didier H. Besset Object-oriented Implementation of Numerical Methods, Morgan-Kaufmann (2001)
    
    Parameters:
    
    z - the z-ordinate to be evaluated
    
    Returns:
    
    the P(Z<=z) for standard normal
  - stdNormalCDF
```
public static double stdNormalCDF(double z)
```
    Computes the cumulative distribution function for a standard normal distribution using Taylor approximation. The approximation is accurate to absolute error less than 8 * 10^(-16). * Reference: Evaluating the Normal Distribution by George Marsaglia. * http://www.jstatsoft.org/v11/a04/paper
    
    Parameters:
    
    z - the z-ordinate to be evaluated
    
    Returns:
    
    the P(Z<=z) for standard normal
  - stdNormalPDF
```
public static double stdNormalPDF(double z)
```
    Computes the pdf function for a standard normal distribution from Abramovitz and Stegun, see also Didier H. Besset Object-oriented Implementation of Numerical Methods, Morgan-Kaufmann (2001)
    
    Parameters:
    
    z - the z-ordinate to be evaluated
    
    Returns:
    
    the f(z) for standard normal
  - stdNormalInvCDF
```
public static double stdNormalInvCDF(double p)
```
    Computes the inverse cumulative distribution function for a standard normal distribution see, W. J. Cody, Rational Chebyshev approximations for the error function Math. Comp. pp 631-638 this is without the extra refinement and has relative error of 1.15e-9 http://www.math.uio.no/~jacklam/notes/invnorm/
    
    Parameters:
    
    p - the probability to be evaluated, p must be within [0,1] p = 0.0 returns Double.NEGATIVE_INFINTITY p = 1.0 returns Double.POSITIVE_INFINITY
    
    Returns:
    
    the "z" value associated with the p
  - stdNormalComplementaryCDF
```
public static double stdNormalComplementaryCDF(double z)
```
    Computes the complementary cumulative probability for the standard normal distribution function for given value of z
    
    Parameters:
    
    z - The value to be evaluated
    
    Returns:
    
    The probability, 1-P{X<=z}
  - stdNormalFirstOrderLossFunction
```
public static double stdNormalFirstOrderLossFunction(double z)
```
    Computes the first order loss function for the standard normal distribution function for given value of x, G1(z) = E[max(Z-z,0)]
    
    Parameters:
    
    z - The value to be evaluated
    
    Returns:
    
    The loss function value, E[max(Z-z,0)]
  - stdNormalSecondOrderLossFunction
```
public static double stdNormalSecondOrderLossFunction(double z)
```
    Computes the 2nd order loss function for the standard normal distribution function for given value of z, G2(z) = (1/2)E[max(Z-z,0)*max(Z-z-1,0)]
    
    Parameters:
    
    z - The value to be evaluated
    
    Returns:
    
    The loss function value, (1/2)E[max(Z-z,0)*max(Z-z-1,0)]
  - cdf
```
public final double cdf(double x)
```
    Description copied from interface: CDFIfc
    
    Returns the F(x) = Pr{X <= x} where F represents the cumulative distribution function
    
    Specified by:
    
    cdf in interface CDFIfc
    
    Parameters:
    
    x - a double representing the upper limit
    
    Returns:
    
    a double representing the probability
  - pdf
```
public final double pdf(double x)
```
    Description copied from interface: PDFIfc
    
    Returns the f(x) where f represents the probability density function for the distribution. Note this is not a probability.
    
    Specified by:
    
    pdf in interface PDFIfc
    
    Parameters:
    
    x - a double representing the value to be evaluated
    
    Returns:
    
    f(x)
  - invCDF
```
public final double invCDF(double p)
```
    Description copied from interface: InverseCDFIfc
    
    Provides the inverse cumulative distribution function for the distribution While closed form solutions for the inverse cdf may not exist, numerical search methods can be used to solve F(X) = U.
    
    Specified by:
    
    invCDF in interface InverseCDFIfc
    
    Parameters:
    
    p - The probability to be evaluated for the inverse, p must be [0,1] or an IllegalArgumentException is thrown
    
    Returns:
    
    The inverse cdf evaluated at the supplied probability
  - getKurtosis
```
public final double getKurtosis()
```
    Gets the kurtosis of the distribution
    
    Returns:
    
    the kurtosis
  - getSkewness
```
public final double getSkewness()
```
    Gets the skewness of the distribution
    
    Returns:
    
    the skewness
  - complementaryCDF
```
public double complementaryCDF(double x)
```
    Description copied from interface: CDFIfc
    
    Computes the complementary cumulative probability distribution function for given value of x
    
    Specified by:
    
    complementaryCDF in interface CDFIfc
    
    Parameters:
    
    x - The value to be evaluated
    
    Returns:
    
    The probability, 1-P{X<=x}
  - firstOrderLossFunction
```
public double firstOrderLossFunction(double x)
```
    Description copied from interface: FirstOrderLossFunctionIfc
    
    Computes the first order loss function for the function for given value of x, G1(x) = E[max(X-x,0)]
    
    Specified by:
    
    firstOrderLossFunction in interface FirstOrderLossFunctionIfc
    
    Parameters:
    
    x - The value to be evaluated
    
    Returns:
    
    The loss function value, E[max(X-x,0)]
  - secondOrderLossFunction
```
public double secondOrderLossFunction(double x)
```
    Description copied from interface: SecondOrderLossFunctionIfc
    
    Computes the 2nd order loss function for the distribution function for given value of x, G2(x) = (1/2)E[max(X-x,0)*max(X-x-1,0)]
    
    Specified by:
    
    secondOrderLossFunction in interface SecondOrderLossFunctionIfc
    
    Parameters:
    
    x - The value to be evaluated
    
    Returns:
    
    The loss function value, (1/2)E[max(X-x,0)*max(X-x-1,0)]
  - setParameters
```
public void setParameters(double[] parameters)
```
    Sets the parameters for the distribution mean = parameters[0] and variance = parameters[1]
    
    Specified by:
    
    setParameters in interface ParametersIfc
    
    Parameters:
    
    parameters - an array of doubles representing the parameters for the distribution
  - getParameters
```
public double[] getParameters()
```
    Gets the parameters for the distribution
    
    Specified by:
    
    getParameters in interface ParametersIfc
    
    Returns:
    
    Returns an array of the parameters for the distribution
  - getRandomVariable
```
public final RVariableIfc getRandomVariable(RNStreamIfc rng)
```
    Specified by:
    
    getRandomVariable in interface GetRVariableIfc
    
    Overrides:
    
    getRandomVariable in class Distribution
    
    Parameters:
    
    rng - the stream to use
    
    Returns:
    
    a random variable

Class Normal

Nested Class Summary

Nested classes/interfaces inherited from class jsl.utilities.distributions.Distribution

Field Summary

Fields inherited from class jsl.utilities.distributions.Distribution

Constructor Summary

Method Summary

Methods inherited from class jsl.utilities.distributions.Distribution

Methods inherited from class java.lang.Object

Methods inherited from interface jsl.utilities.distributions.CDFIfc

Methods inherited from interface jsl.utilities.random.rvariable.GetRVariableIfc

Methods inherited from interface jsl.utilities.distributions.VarianceIfc

Constructor Detail

Normal

Normal

Normal

Normal

Method Detail

newInstance

getDomain

setMean

getMean

setVariance

getVariance

stdNormalCDFAbramovitzAndStegun

stdNormalCDF

stdNormalPDF

stdNormalInvCDF

stdNormalComplementaryCDF

stdNormalFirstOrderLossFunction

stdNormalSecondOrderLossFunction

cdf

pdf

invCDF

getKurtosis

getSkewness

complementaryCDF

firstOrderLossFunction

secondOrderLossFunction

setParameters

getParameters

getRandomVariable