Class PascalDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractDiscreteDistribution
org.apache.commons.statistics.distribution.PascalDistribution
All Implemented Interfaces:
DiscreteDistribution
public final class PascalDistribution extends AbstractDiscreteDistribution
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:

  • Field Details

    • numberOfSuccesses

      private final int numberOfSuccesses
      The number of successes.

    • probabilityOfSuccess

      private final double probabilityOfSuccess
      The probability of success.

    • logProbabilityOfSuccessByNumOfSuccesses

      private final double logProbabilityOfSuccessByNumOfSuccesses
      The value of log(p) * n, where p is the probability of success and n is the number of successes, stored for faster computation.

    • log1mProbabilityOfSuccess

      private final double log1mProbabilityOfSuccess
      The value of log(1-p), where p is the probability of success, stored for faster computation.

    • probabilityOfSuccessPowNumOfSuccesses

      private final double probabilityOfSuccessPowNumOfSuccesses
      The value of p^n, where p is the probability of success and n is the number of successes, stored for faster computation.

  • Constructor Details

    • PascalDistribution

      private PascalDistribution(int r, double p)
      Parameters:
      r - Number of successes.
      p - Probability of success.

  • 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.