## 15.1 Negative binomial distribution

For the negative binomial distribution Stan uses the parameterization described in Gelman et al. (2013). For alternative parameterizations, see section negative binomial glm.

### 15.1.1 Probability mass function

If $$\alpha \in \mathbb{R}^+$$ and $$\beta \in \mathbb{R}^+$$, then for $$n \in \mathbb{N}$$, $\text{NegBinomial}(n~|~\alpha,\beta) = \binom{n + \alpha - 1}{\alpha - 1} \, \left( \frac{\beta}{\beta+1} \right)^{\!\alpha} \, \left( \frac{1}{\beta + 1} \right)^{\!n} \!.$

The mean and variance of a random variable $$n \sim \text{NegBinomial}(\alpha,\beta)$$ are given by $\mathbb{E}[n] = \frac{\alpha}{\beta} \ \ \text{ and } \ \ \text{Var}[n] = \frac{\alpha}{\beta^2} (\beta + 1).$

### 15.1.2 Sampling statement

n ~ neg_binomial(alpha, beta)

Increment target log probability density with neg_binomial_lupmf(n | alpha, beta).
Available since 2.0

### 15.1.3 Stan functions

real neg_binomial_lpmf(ints n | reals alpha, reals beta)
The log negative binomial probability mass of n given shape alpha and inverse scale beta
Available since 2.12

real neg_binomial_lupmf(ints n | reals alpha, reals beta)
The log negative binomial probability mass of n given shape alpha and inverse scale beta dropping constant additive terms
Available since 2.25

real neg_binomial_cdf(ints n, reals alpha, reals beta)
The negative binomial cumulative distribution function of n given shape alpha and inverse scale beta
Available since 2.0

real neg_binomial_lcdf(ints n | reals alpha, reals beta)
The log of the negative binomial cumulative distribution function of n given shape alpha and inverse scale beta
Available since 2.12

real neg_binomial_lccdf(ints n | reals alpha, reals beta)
The log of the negative binomial complementary cumulative distribution function of n given shape alpha and inverse scale beta
Available since 2.12

R neg_binomial_rng(reals alpha, reals beta)
Generate a negative binomial variate with shape alpha and inverse scale beta; may only be used in transformed data and generated quantities blocks. alpha $$/$$ beta must be less than $$2 ^ {29}$$. For a description of argument and return types, see section vectorized function signatures.
Available since 2.18

### References

Gelman, Andrew, J. B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. Third Edition. London: Chapman & Hall / CRC Press.