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 1535290 2013-10-24 06:58:32Z luc $
  See Also:
    Normal distribution (Wikipedia), Normal distribution (MathWorld), Serialized Form

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static double` | `DEFAULT_INVERSE_ABSOLUTE_ACCURACY` Default inverse cumulative probability accuracy.

    - ### 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)` Creates a normal distribution.
    `NormalDistribution(RandomGenerator rng, double mean, double sd, double inverseCumAccuracy)` Creates a normal distribution.

  - ### Method Summary

    Methods
    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` | `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.
    `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.
    `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` | `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)`.
    `double` | `sample()` Generate a random value sampled from this distribution.

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

      `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

  - ### 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)
                             throws NotStrictlyPositiveException

      Creates a normal distribution.
      Parameters:
        `rng` - Random number generator.
        `mean` - Mean for this distribution.
        `sd` - Standard deviation for this distribution.
      Throws:
        `NotStrictlyPositiveException` - if `sd <= 0`.
      Since:
        3.3

    - #### 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`

    - #### 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)`.
      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 OutOfRangeException

      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:
        `OutOfRangeException` - if `p < 0` or `p > 1`
      Since:
        3.2

    - #### 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.

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