Package org.apache.commons.statistics.distribution

Class PascalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.PascalDistribution
All Implemented Interfaces:
DiscreteDistribution
public final class PascalDistribution extends Object
Implementation of the Pascal distribution.

The Pascal distribution is a special case of the negative binomial distribution where the number of successes parameter is an integer.

There are various ways to express the probability mass and distribution functions for the Pascal distribution. The present implementation represents the distribution of the number of failures before \( r \) successes occur. This is the convention adopted in e.g. MathWorld, but not in Wikipedia.

The probability mass function of \( X \) is:

\[ f(k; r, p) = \binom{k+r-1}{r-1} p^r \, (1-p)^k \]

for \( r \in \{1, 2, \dots\} \) the number of successes, \( p \in (0, 1] \) the probability of success, \( k \in \{0, 1, 2, \dots\} \) the total number of failures, and

\[ \binom{k+r-1}{r-1} = \frac{(k+r-1)!}{(r-1)! \, k!} \]

is the binomial coefficient.

The cumulative distribution function of \( X \) is:

\[ P(X \leq k) = I(p, r, k + 1) \]

where \( I \) is the regularized incomplete beta function.

See Also:

  • Method Details

    • of

      public static PascalDistribution of(int r, double p)
      Create a Pascal distribution.
      Parameters:
      r - Number of successes.
      p - Probability of success.
      Returns:
      the distribution
      Throws:
      IllegalArgumentException - if r <= 0 or p <= 0 or p > 1.

    • getNumberOfSuccesses

      public int getNumberOfSuccesses()
      Gets the number of successes parameter of this distribution.
      Returns:
      the number of successes.

    • getProbabilityOfSuccess

      public double getProbabilityOfSuccess()
      Gets the probability of success parameter of this distribution.
      Returns:
      the probability of success.

    • probability

      public double probability(int 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 probability mass function (PMF) for the distribution.
      Parameters:
      x - Point at which the PMF is evaluated.
      Returns:
      the value of the probability mass function at x.

    • logProbability

      public double logProbability(int x)
      For a random variable X whose values are distributed according to this distribution, this method returns log(P(X = x)), where log is the natural logarithm.
      Parameters:
      x - Point at which the PMF is evaluated.
      Returns:
      the logarithm of the value of the probability mass function at x.

    • cumulativeProbability

      public double cumulativeProbability(int 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(int 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.

    • getMean

      public double getMean()
      Gets the mean of this distribution.

      For number of successes \( r \) and probability of success \( p \), the mean is:

      \[ \frac{r (1 - p)}{p} \]

      Returns:
      the mean.

    • getVariance

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

      For number of successes \( r \) and probability of success \( p \), the variance is:

      \[ \frac{r (1 - p)}{p^2} \]

      Returns:
      the variance.

    • getSupportLowerBound

      public int getSupportLowerBound()
      Gets the lower bound of the support. This method must return the same value as inverseCumulativeProbability(0), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) \gt 0 \} \). By convention, Integer.MIN_VALUE should be substituted for negative infinity.

      The lower bound of the support is always 0.

      Returns:
      0.

    • getSupportUpperBound

      public int getSupportUpperBound()
      Gets the upper bound of the support. This method must return the same value as inverseCumulativeProbability(1), i.e. \( \inf \{ x \in \mathbb Z : P(X \le x) = 1 \} \). By convention, Integer.MAX_VALUE should be substituted for positive infinity.

      The upper bound of the support is positive infinity except for the probability parameter p = 1.0.

      Returns:
      Integer.MAX_VALUE or 0.

    • probability

      public double probability(int x0, int 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)

      Special cases:

      • returns 0.0 if x0 == x1;
      • returns probability(x1) if x0 + 1 == x1;
      Specified by:
      probability in interface DiscreteDistribution
      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.

    • inverseCumulativeProbability

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

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

      If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of cumulativeProbability(x) called using the returned p-quantile may not compute the original p.

      The default implementation returns:

      Specified by:
      inverseCumulativeProbability in interface DiscreteDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest p-quantile of this distribution (largest 0-quantile for p = 0).
      Throws:
      IllegalArgumentException - if p < 0 or p > 1

    • inverseSurvivalProbability

      public int 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:

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

      If the result exceeds the range of the data type int, then Integer.MIN_VALUE or Integer.MAX_VALUE is returned. In this case the result of survivalProbability(x) called using the returned (1-p)-quantile may not compute the original p.

      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 DiscreteDistribution
      Parameters:
      p - Cumulative probability.
      Returns:
      the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).
      Throws:
      IllegalArgumentException - if p < 0 or p > 1

    • createSampler

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