Class NormalDistribution

    • Field Summary

      Fields 
      Modifier and Type Field Description
      private double logStandardDeviationPlusHalfLog2Pi
      The value of log(sd) + 0.5*log(2*pi) stored for faster computation.
      private double mean
      Mean of this distribution.
      private double sdSqrt2
      Standard deviation multiplied by sqrt(2).
      private double sdSqrt2pi
      Standard deviation multiplied by sqrt(2 pi).
      private double standardDeviation
      Standard deviation of this distribution.
    • Constructor Summary

      Constructors 
      Modifier Constructor Description
      private NormalDistribution​(double mean, double sd)  
    • Method Summary

      All Methods Static Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      ContinuousDistribution.Sampler createSampler​(org.apache.commons.rng.UniformRandomProvider rng)
      Creates a sampler.
      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()
      Gets the mean of this distribution.
      double getStandardDeviation()
      Gets the standard deviation parameter of this distribution.
      double getSupportLowerBound()
      Gets the lower bound of the support.
      double getSupportUpperBound()
      Gets the upper bound of the support.
      double getVariance()
      Gets the variance of this distribution.
      double inverseCumulativeProbability​(double p)
      Computes the quantile function of this distribution.
      double inverseSurvivalProbability​(double p)
      Computes the inverse survival probability function of this distribution.
      double logDensity​(double x)
      Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
      static NormalDistribution of​(double mean, double sd)
      Creates a normal distribution.
      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 survivalProbability​(double x)
      For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).
      • Methods inherited from class java.lang.Object

        clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
    • Field Detail

      • mean

        private final double mean
        Mean of this distribution.
      • standardDeviation

        private final double standardDeviation
        Standard deviation of this distribution.
      • logStandardDeviationPlusHalfLog2Pi

        private final double logStandardDeviationPlusHalfLog2Pi
        The value of log(sd) + 0.5*log(2*pi) stored for faster computation.
      • sdSqrt2

        private final double sdSqrt2
        Standard deviation multiplied by sqrt(2). This is used to avoid a double division when computing the value passed to the error function:
          ((x - u) / sd) / sqrt(2) == (x - u) / (sd * sqrt(2)).
          

        Note: Implementations may first normalise x and then divide by sqrt(2) resulting in differences due to rounding error that show increasingly large relative differences as the error function computes close to 0 in the extreme tail.

      • sdSqrt2pi

        private final double sdSqrt2pi
        Standard deviation multiplied by sqrt(2 pi). Computed to high precision.
    • Constructor Detail

      • NormalDistribution

        private NormalDistribution​(double mean,
                                   double sd)
        Parameters:
        mean - Mean for this distribution.
        sd - Standard deviation for this distribution.
    • Method Detail

      • of

        public static NormalDistribution of​(double mean,
                                            double sd)
        Creates a normal distribution.
        Parameters:
        mean - Mean for this distribution.
        sd - Standard deviation for this distribution.
        Returns:
        the distribution
        Throws:
        java.lang.IllegalArgumentException - if sd <= 0.
      • getStandardDeviation

        public double getStandardDeviation()
        Gets the standard deviation parameter of this distribution.
        Returns:
        the standard deviation.
      • 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 - Point at which the PDF is evaluated.
        Returns:
        the value of the probability density function at x.
      • probability

        public 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). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)
        Specified by:
        probability in interface ContinuousDistribution
        Overrides:
        probability in class AbstractContinuousDistribution
        Parameters:
        x0 - Lower bound (exclusive).
        x1 - Upper bound (inclusive).
        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.
      • 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.
        Parameters:
        x - Point at which the PDF is evaluated.
        Returns:
        the logarithm of the value of the probability density function at 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.
        Parameters:
        x - 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.
      • survivalProbability

        public double survivalProbability​(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 complementary cumulative distribution function.

        By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

        Parameters:
        x - Point at which the survival function is evaluated.
        Returns:
        the probability that a random variable with this distribution takes a value greater than x.
      • inverseSurvivalProbability

        public double inverseSurvivalProbability​(double p)
        Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

        \[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

        By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

        The default implementation returns:

        Specified by:
        inverseSurvivalProbability in interface ContinuousDistribution
        Overrides:
        inverseSurvivalProbability in class AbstractContinuousDistribution
        Parameters:
        p - Survival probability.
        Returns:
        the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
      • getMean

        public double getMean()
        Gets the mean of this distribution.
        Returns:
        the mean.
      • getVariance

        public double getVariance()
        Gets the variance of this distribution.

        For standard deviation parameter \( \sigma \), the variance is \( \sigma^2 \).

        Returns:
        the variance.
      • getSupportLowerBound

        public double getSupportLowerBound()
        Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

        The lower bound of the support is always negative infinity.

        Returns:
        negative infinity.
      • getSupportUpperBound

        public double getSupportUpperBound()
        Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb R : P(X \le x) = 1 \} \).

        The upper bound of the support is always positive infinity.

        Returns:
        positive infinity.