Class TDistribution.NormalTDistribution

All Implemented Interfaces:
ContinuousDistribution
Enclosing class:
TDistribution

private static class TDistribution.NormalTDistribution extends TDistribution
Specialisation of the T-distribution used when there are infinite degrees of freedom. In this case the distribution matches a normal distribution. This is used when the variance is not different from 1.0.

This delegates all methods to the standard normal distribution. Instances are allowed to provide access to the degrees of freedom used during construction.

  • Constructor Details

    • NormalTDistribution

      NormalTDistribution(double degreesOfFreedom)
      Parameters:
      degreesOfFreedom - Degrees of freedom.
  • Method Details

    • density

      public double density(double x)
      Description copied from interface: ContinuousDistribution
      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)
      Description copied from class: AbstractContinuousDistribution
      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)
      Description copied from interface: ContinuousDistribution
      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)
      Description copied from interface: ContinuousDistribution
      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.
    • inverseCumulativeProbability

      public double inverseCumulativeProbability(double p)
      Description copied from class: AbstractContinuousDistribution
      Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

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

      The default implementation returns:

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

      public double getMean()
      Description copied from class: TDistribution
      Gets the mean of this distribution.

      For degrees of freedom parameter \( v \), the mean is:

      \[ \mathbb{E}[X] = \begin{cases} 0 & \text{for } v \gt 1 \\ \text{undefined} & \text{otherwise} \end{cases} \]

      Specified by:
      getMean in interface ContinuousDistribution
      Specified by:
      getMean in class TDistribution
      Returns:
      the mean, or NaN if it is not defined.
    • getVariance

      public double getVariance()
      Description copied from class: TDistribution
      Gets the variance of this distribution.

      For degrees of freedom parameter \( v \), the variance is:

      \[ \operatorname{var}[X] = \begin{cases} \frac{v}{v - 2} & \text{for } v \gt 2 \\ \infty & \text{for } 1 \lt v \le 2 \\ \text{undefined} & \text{otherwise} \end{cases} \]

      Specified by:
      getVariance in interface ContinuousDistribution
      Specified by:
      getVariance in class TDistribution
      Returns:
      the variance, or NaN if it is not defined.
    • createSampler

      public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)
      Description copied from class: AbstractContinuousDistribution
      Creates a sampler.
      Specified by:
      createSampler in interface ContinuousDistribution
      Overrides:
      createSampler in class AbstractContinuousDistribution
      Parameters:
      rng - Generator of uniformly distributed numbers.
      Returns:
      a sampler that produces random numbers according this distribution.