# Negative binomial distribution

**negative binomialGamma-Poisson distributioninverse binomial distributionnegative binomial random variablenegative binomial regressionnegative-binomialnegative-binomially distributedPascalPolya distributionthe one for the probability mass function when ''r'' is real-valued**

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.wikipedia

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### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.

Well-known discrete probability distributions used in statistical modeling include the Poisson distribution, the Bernoulli distribution, the binomial distribution, the geometric distribution, and the negative binomial distribution.

### Poisson distribution

**PoissonPoisson-distributedPoissonian**

For occurrences of "contagious" discrete events, like tornado outbreaks, the Polya distributions can be used to give more accurate models than the Poisson distribution by allowing the mean and variance to be different, unlike the Poisson. The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

The posterior predictive distribution for a single additional observation is a negative binomial distribution, sometimes called a Gamma–Poisson distribution.

### Bernoulli process

**BernoulliBernoulli sequenceBernoulli variable**

In other words, the negative binomial distribution is the probability distribution of the number of successes before the rth failure in a Bernoulli process, with probability p of successes on each trial.

The negative binomial variables may be interpreted as random waiting times.

### Probability theory

**theory of probabilityprobabilityprobability theorist**

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.

Some fundamental discrete distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions.

### Bernoulli trial

**Bernoulli trialsBernoulli random variablesBernoulli-distributed**

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted r) occurs.

Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions.

### Cumulant

**cumulant generating functioncumulant-generating functioncumulants**

See Cumulants of some discrete probability distributions.

* The negative binomial distributions, (number of failures before n successes with probability p of success on each trial).

### George Pólya

**PólyaGeorge PolyaGyörgy Pólya**

The Pascal distribution (after Blaise Pascal) and Polya distribution (for George Pólya) are special cases of the negative binomial distribution.

### Overdispersion

**underdispersionover-dispersionoverdispersed**

In such cases, the observations are overdispersed with respect to a Poisson distribution, for which the mean is equal to the variance.

In the case of count data, a Poisson mixture model like the negative binomial distribution can be proposed instead, in which the mean of the Poisson distribution can itself be thought of as a random variable drawn – in this case – from the gamma distribution thereby introducing an additional free parameter (note the resulting negative binomial distribution is completely characterized by two parameters).

### Poisson regression

**PoissonDiscrete RegressionNegative binomial regression**

Under some circumstances, the problem of overdispersion can be solved by using quasi-likelihood estimation or a negative binomial distribution instead.

### Compound probability distribution

**compound distributionmixturecompounding**

The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

The distribution may be generalized by allowing for variability in its rate parameter, implemented via a gamma distribution, which results in a marginal negative binomial distribution.

### Infinite divisibility (probability)

**infinitely divisibleinfinite divisibilityinfinitely divisible distribution**

The negative binomial distribution is infinitely divisible, i.e., if Y has a negative binomial distribution, then for any positive integer n, there exist independent identically distributed random variables Y 1, ..., Y n whose sum has the same distribution that Y has.

The Poisson distribution, the negative binomial distribution, the Gamma distribution and the degenerate distribution are examples of infinitely divisible distributions; as are the normal distribution, Cauchy distribution and all other members of the stable distribution family.

### Geometric distribution

**geometricgeometrically distributed geometrically distributed**

the probability of the first failure occurring on the (k + 1)st trial), which is a geometric distribution:

* The geometric distribution Y is a special case of the negative binomial distribution, with r = 1.

### Compound Poisson distribution

**compound Poisson-gamma distributioncompound Poisson–gammacompound Poisson–gamma distribution**

The negative binomial distribution NB(r,p) can be represented as a compound Poisson distribution: Let {Y n, n ∈ ℕ 0 } denote a sequence of independent and identically distributed random variables, each one having the logarithmic distribution Log(p), with probability mass function

Other special cases include: shiftgeometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria–Delbrück distribution in Luria–Delbrück experiment.

### Gamma distribution

**gammagamma distributedGamma variate**

The negative binomial distribution also arises as a continuous mixture of Poisson distributions (i.e. a compound probability distribution) where the mixing distribution of the Poisson rate is a gamma distribution.

The gamma distribution is also used to model errors in multi-level Poisson regression models, because the combination of the Poisson distribution and a gamma distribution is a negative binomial distribution.

### Binomial distribution

**binomialbinomial modelBinomial probability**

Furthermore, if B s+r is a random variable following the binomial distribution with parameters s + r and 1 − p, then

### Discrete phase-type distribution

**discrete phase-type distributed**

### Hypergeometric distribution

**multivariate hypergeometric distributionhypergeometrichypergeometric test**

### Logarithmic distribution

**Fisher Log-Serieslogarithmic (series) distributionlogarithmic series distribution**

The negative binomial distribution NB(r,p) can be represented as a compound Poisson distribution: Let {Y n, n ∈ ℕ 0 } denote a sequence of independent and identically distributed random variables, each one having the logarithmic distribution Log(p), with probability mass function

A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution.

### Negative multinomial distribution

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(r, p)) to more than two outcomes.

### Exponential family

**exponential familiesnatural parameternatural parameters**

The family of negative binomial distributions with fixed number of failures (a.k.a. stopping-time parameter) r is an exponential family.

### Extended negative binomial distribution

In probability and statistics the extended negative binomial distribution is a discrete probability distribution extending the negative binomial distribution.

### Beta negative binomial distribution

**(beta-negative binomial)**

In the above, NB(r, p) is the negative binomial distribution and B(α, β) is the beta distribution.

### Binomial theorem

**binomial expansionbinomial formulabinomial**

Using Newton's binomial theorem, this can equally be written as:

The binomial theorem is closely related to the probability mass function of the negative binomial distribution.

### Beta function

**regularized incomplete beta functionincomplete beta functionEuler beta function**

The cumulative distribution function can be expressed in terms of the regularized incomplete beta function: