org.hipparchus.distribution.continuous

Class NormalDistribution

java.lang.Object

org.hipparchus.distribution.continuous.AbstractRealDistribution

org.hipparchus.distribution.continuous.NormalDistribution

All Implemented Interfaces:

Serializable, RealDistribution

public class NormalDistribution
extends AbstractRealDistribution

Implementation of the normal (gaussian) distribution.

See Also:

Normal distribution (Wikipedia), Normal distribution (MathWorld), Serialized Form

Field Summary

Fields inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

DEFAULT_SOLVER_ABSOLUTE_ACCURACY

Constructor Summary

Constructors

Constructor and Description
NormalDistribution() Create a normal distribution with mean equal to zero and standard deviation equal to one.
NormalDistribution(double mean, double sd) Create a normal distribution using the given mean, standard deviation.

Method Summary

Modifier and Type	Method and Description
double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getMean() Access the mean.
double	getNumericalMean() Use this method to get the numerical value of the mean of this distribution.
double	getNumericalVariance() Use this method to get the numerical value of the variance of this distribution.
double	getStandardDeviation() Access the standard deviation.
double	getSupportLowerBound() Access the lower bound of the support.
double	getSupportUpperBound() Access the upper bound of the support.
double	inverseCumulativeProbability(double p) Computes the quantile function of this distribution.
boolean	isSupportConnected() Use this method to get information about whether the support is connected, i.e.
double	logDensity(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
double	probability(double x0, double x1) For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).

Methods inherited from class org.hipparchus.distribution.continuous.AbstractRealDistribution

getSolverAbsoluteAccuracy

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

NormalDistribution

public NormalDistribution()

Create a normal distribution with mean equal to zero and standard deviation equal to one.

NormalDistribution

public NormalDistribution(double mean,
                          double sd)
                   throws MathIllegalArgumentException

Create a normal distribution using the given mean, standard deviation.

Parameters:

mean - Mean for this distribution.

sd - Standard deviation for this distribution.

Throws:

MathIllegalArgumentException - if sd <= 0.

Method Detail

getMean

public double getMean()

Access the mean.

Returns:

the mean for this distribution.

getStandardDeviation

public double getStandardDeviation()

Access the standard deviation.

Returns:

the standard deviation for this distribution.

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Parameters:

x - the point at which the PDF is evaluated

Returns:

the value of the probability density function at point x

logDensity

public double logDensity(double x)

Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient. Note that due to the floating point precision and under/overflow issues, this method will for some distributions be more precise and faster than computing the logarithm of RealDistribution.density(double).

The default implementation simply computes the logarithm of density(x).

Specified by:

logDensity in interface RealDistribution

Overrides:

logDensity in class AbstractRealDistribution

Parameters:

x - the point at which the PDF is evaluated

Returns:

the logarithm of the value of the probability density function at point x

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution. If x is more than 40 standard deviations from the mean, 0 or 1 is returned, as in these cases the actual value is within Double.MIN_VALUE of 0 or 1.

Parameters:

x - the point at which the CDF is evaluated

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)
                                    throws MathIllegalArgumentException

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is

• inf{x in R | P(X<=x) >= p} for 0 < p <= 1,
• inf{x in R | P(X<=x) > 0} for p = 0.

The default implementation returns

• RealDistribution.getSupportLowerBound() for p = 0,
• RealDistribution.getSupportUpperBound() for p = 1.

Specified by:

inverseCumulativeProbability in interface RealDistribution

Overrides:

inverseCumulativeProbability in class AbstractRealDistribution

Parameters:

p - the cumulative probability

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0)

Throws:

MathIllegalArgumentException - if p < 0 or p > 1

probability

public double probability(double x0,
                          double x1)
                   throws MathIllegalArgumentException

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).

Specified by:

probability in interface RealDistribution

Overrides:

probability in class AbstractRealDistribution

Parameters:

x0 - Lower bound (excluded).

x1 - Upper bound (included).

Returns:

the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.

Throws:

MathIllegalArgumentException - if x0 > x1. The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

getNumericalMean

public double getNumericalMean()

Use this method to get the numerical value of the mean of this distribution. For mean parameter mu, the mean is mu.

Returns:

the mean or Double.NaN if it is not defined

getNumericalVariance

public double getNumericalVariance()

Use this method to get the numerical value of the variance of this distribution. For standard deviation parameter s, the variance is s^2.

Returns:

the variance (possibly Double.POSITIVE_INFINITY as for certain cases in TDistribution) or Double.NaN if it is not defined

getSupportLowerBound

public double getSupportLowerBound()

Access the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0). In other words, this method must return

inf {x in R | P(X <= x) > 0}.

The lower bound of the support is always negative infinity no matter the parameters.

Returns:

lower bound of the support (always Double.NEGATIVE_INFINITY)

getSupportUpperBound

public double getSupportUpperBound()

Access the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1). In other words, this method must return

inf {x in R | P(X <= x) = 1}.

The upper bound of the support is always positive infinity no matter the parameters.

Returns:

upper bound of the support (always Double.POSITIVE_INFINITY)

isSupportConnected

public boolean isSupportConnected()

Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:

true