**'org.apache.commons.math3.distribution.PascalDistribution'**Java class

## Class PascalDistribution

- java.lang.Object
- org.apache.commons.math3.distribution.AbstractIntegerDistribution
- org.apache.commons.math3.distribution.PascalDistribution

- All Implemented Interfaces:
- Serializable, IntegerDistribution

public class PascalDistributionextends AbstractIntegerDistribution

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.

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