org.apache.commons.math3.distribution

## Class NormalDistribution

- java.lang.Object
- - org.apache.commons.math3.distribution.AbstractRealDistribution
  - - org.apache.commons.math3.distribution.NormalDistribution

All Implemented Interfaces:
  Serializable, RealDistribution

    public class NormalDistribution
    extends AbstractRealDistribution

Implementation of the normal (gaussian) distribution.

Version:
  $Id: NormalDistribution.java 1416643 2012-12-03 19:37:14Z tn $
See Also:
  Normal distribution (Wikipedia), Normal distribution (MathWorld), Serialized Form

### Field Summary

Fields
`static double` `DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.
`private double` `mean` Mean of this distribution.
`private static long` `serialVersionUID` Serializable version identifier.
`private double` `solverAbsoluteAccuracy` Inverse cumulative probability accuracy.
`private static double` `SQRT2` √(2)
`private static double` `SQRT2PI` √(2 π)
`private double` `standardDeviation` Standard deviation of this distribution.

### Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

`random, randomData, SOLVER_DEFAULT_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 and standard deviation.
`NormalDistribution(double mean, double sd, double inverseCumAccuracy)` Create a normal distribution using the given mean, standard deviation and inverse cumulative distribution accuracy.
`NormalDistribution(RandomGenerator rng, double mean, double sd, double inverseCumAccuracy)` Creates a normal distribution.

### Method Summary

Methods
`double` `cumulativeProbability(double x)` For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
`double` `cumulativeProbability(double x0, double x1)` Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
`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.
`protected double` `getSolverAbsoluteAccuracy()` Returns the solver absolute accuracy for inverse cumulative computation.
`double` `getStandardDeviation()` Access the standard deviation.
`double` `getSupportLowerBound()` Access the lower bound of the support.
`double` `getSupportUpperBound()` Access the upper bound of the support.
`boolean` `isSupportConnected()` Use this method to get information about whether the support is connected, i.e.
`boolean` `isSupportLowerBoundInclusive()` Whether or not the lower bound of support is in the domain of the density function.
`boolean` `isSupportUpperBoundInclusive()` Whether or not the upper bound of support is in the domain of the density function.
`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)`.
`double` `sample()` Generate a random value sampled from this distribution.

### Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution

`inverseCumulativeProbability, probability, reseedRandomGenerator, sample`

### Methods inherited from class java.lang.Object

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

### Field Detail

#### DEFAULT_INVERSE_ABSOLUTE_ACCURACY

    public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY

Default inverse cumulative probability accuracy.

Since:
  2.1
See Also:
  Constant Field Values

#### serialVersionUID

    private static final long serialVersionUID

Serializable version identifier.

See Also:
  Constant Field Values

#### SQRT2PI

    private static final double SQRT2PI

√(2 π)

#### SQRT2

    private static final double SQRT2

√(2)

#### mean

    private final double mean

Mean of this distribution.

#### standardDeviation

    private final double standardDeviation

Standard deviation of this distribution.

#### solverAbsoluteAccuracy

    private final double solverAbsoluteAccuracy

Inverse cumulative probability accuracy.

### 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 NotStrictlyPositiveException

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

Parameters:
  `mean` - Mean for this distribution.
  `sd` - Standard deviation for this distribution.
Throws:
  `NotStrictlyPositiveException` - if `sd <= 0`.

#### NormalDistribution

    public NormalDistribution(double mean,
                      double sd,
                      double inverseCumAccuracy)
                       throws NotStrictlyPositiveException

Create a normal distribution using the given mean, standard deviation and inverse cumulative distribution accuracy.

Parameters:
  `mean` - Mean for this distribution.
  `sd` - Standard deviation for this distribution.
  `inverseCumAccuracy` - Inverse cumulative probability accuracy.
Throws:
  `NotStrictlyPositiveException` - if `sd <= 0`.
Since:
  2.1

#### NormalDistribution

    public NormalDistribution(RandomGenerator rng,
                      double mean,
                      double sd,
                      double inverseCumAccuracy)
                       throws NotStrictlyPositiveException

Creates a normal distribution.

Parameters:
  `rng` - Random number generator.
  `mean` - Mean for this distribution.
  `sd` - Standard deviation for this distribution.
  `inverseCumAccuracy` - Inverse cumulative probability accuracy.
Throws:
  `NotStrictlyPositiveException` - if `sd <= 0`.
Since:
  3.1

### 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`

#### 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`

#### cumulativeProbability

    @Deprecated
    public double cumulativeProbability(double x0,
                                          double x1)
                                 throws NumberIsTooLargeException

Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(x0 < X <= x1)`. The default implementation uses the identity

`P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)`

Specified by:
  `cumulativeProbability` in interface `RealDistribution`
Overrides:
  `cumulativeProbability` in class `AbstractRealDistribution`
Parameters:
  `x0` - the exclusive lower bound
  `x1` - the inclusive upper bound
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:
  `NumberIsTooLargeException` - if `x0 > x1`

#### probability

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

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

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

#### getSolverAbsoluteAccuracy

    protected double getSolverAbsoluteAccuracy()

Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.

Overrides:
  `getSolverAbsoluteAccuracy` in class `AbstractRealDistribution`
Returns:
  the maximum absolute error in inverse cumulative probability estimates

#### 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`)

#### isSupportLowerBoundInclusive

    public boolean isSupportLowerBoundInclusive()

Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value.

Returns:
  true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

#### isSupportUpperBoundInclusive

    public boolean isSupportUpperBoundInclusive()

Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value.

Returns:
  true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

#### 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`

#### sample

    public double sample()

Generate a random value sampled from this distribution. The default implementation uses the inversion method.

Specified by:
  `sample` in interface `RealDistribution`
Overrides:
  `sample` in class `AbstractRealDistribution`
Returns:
  a random value.