org.apache.commons.statistics.distribution

## Class PascalDistribution

• java.lang.Object
• org.apache.commons.statistics.distribution.PascalDistribution
• All Implemented Interfaces:
DiscreteDistribution

```public 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. For a random variable `X` whose values are distributed according to this distribution, the probability mass function is given by
`P(X = k) = C(k + r - 1, r - 1) * p^r * (1 - p)^k,`
where `r` is the number of successes, `p` is the probability of success, and `X` is the total number of failures. `C(n, k)` is the binomial coefficient (`n` choose `k`). The mean and variance of `X` are
`E(X) = (1 - p) * r / p, var(X) = (1 - p) * r / p^2.`
Finally, the cumulative distribution function is given by
`P(X <= k) = I(p, r, k + 1)`, where I is the regularized incomplete Beta function.

• ### Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.DiscreteDistribution

`DiscreteDistribution.Sampler`
• ### Constructor Summary

Constructors
Constructor and Description
```PascalDistribution(int r, double p)```
Create a Pascal distribution with the given number of successes and probability of success.
• ### Method Summary

All Methods
Modifier and Type Method and Description
`DiscreteDistribution.Sampler` `createSampler(org.apache.commons.rng.UniformRandomProvider rng)`
Creates a sampler.
`double` `cumulativeProbability(int x)`
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
`double` `getMean()`
Gets the mean of this distribution.
`int` `getNumberOfSuccesses()`
Access the number of successes for this distribution.
`double` `getProbabilityOfSuccess()`
Access the probability of success for this distribution.
`int` `getSupportLowerBound()`
Gets the lower bound of the support.
`int` `getSupportUpperBound()`
Gets the upper bound of the support.
`double` `getVariance()`
Gets the variance of this distribution.
`int` `inverseCumulativeProbability(double p)`
Computes the quantile function of this distribution.
`boolean` `isSupportConnected()`
Indicates whether the support is connected, i.e.
`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.
`double` `probability(int x)`
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X = x)`.
`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)`.
`static int[]` ```sample(int n, DiscreteDistribution.Sampler sampler)```
Utility function for allocating an array and filling it with `n` samples generated by the given `sampler`.
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Constructor Detail

• #### PascalDistribution

```public PascalDistribution(int r,
double p)```
Create a Pascal distribution with the given number of successes and probability of success.
Parameters:
`r` - Number of successes.
`p` - Probability of success.
Throws:
`IllegalArgumentException` - if `r <= 0` or `p < 0` or `p > 1`.
• ### Method Detail

• #### getNumberOfSuccesses

`public int getNumberOfSuccesses()`
Access the number of successes for this distribution.
Returns:
the number of successes.
• #### getProbabilityOfSuccess

`public double getProbabilityOfSuccess()`
Access the probability of success for 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`.
• #### getMean

`public double getMean()`
Gets the mean of this distribution. For number of successes `r` and probability of success `p`, the mean is `r * (1 - p) / p`.
Returns:
the mean, or `Double.NaN` if it is not defined.
• #### getVariance

`public double getVariance()`
Gets the variance of this distribution. For number of successes `r` and probability of success `p`, the variance is `r * (1 - p) / p^2`.
Returns:
the variance, or `Double.NaN` if it is not defined.
• #### 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 Z | P(X <= x) > 0}`. By convention, `Integer.MIN_VALUE` should be substituted for negative infinity. The lower bound of the support is always 0 no matter the parameters.
Returns:
lower bound of the support (always 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 R | P(X <= x) = 1}`. By convention, `Integer.MAX_VALUE` should be substituted for positive infinity. The upper bound of the support is always positive infinity no matter the parameters. Positive infinity is symbolized by `Integer.MAX_VALUE`.
Returns:
upper bound of the support (always `Integer.MAX_VALUE` for positive infinity)
• #### isSupportConnected

`public boolean isSupportConnected()`
Indicates whether the support is connected, i.e. whether all integers between the lower and upper bound of the support are included in the support. The support of this distribution is connected.
Returns:
`true`
• #### 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)`
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 will take 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
• `inf{x in Z | P(X<=x) >= p}` for `0 < p <= 1`,
• `inf{x in Z | P(X<=x) > 0}` for `p = 0`.
If the result exceeds the range of the data type `int`, then `Integer.MIN_VALUE` or `Integer.MAX_VALUE` is returned. 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`).
• #### sample

```public static int[] sample(int n,
DiscreteDistribution.Sampler sampler)```
Utility function for allocating an array and filling it with `n` samples generated by the given `sampler`.
Parameters:
`n` - Number of samples.
`sampler` - Sampler.
Returns:
an array of size `n`.
• #### 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.