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## 13.1 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.

### 13.1.1 Probability Mass Function

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

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

### 13.1.2 Sampling Statement

n ~ neg_binomial(alpha, beta)

Increment target log probability density with neg_binomial_lpmf( n | alpha, beta) dropping constant additive terms.

### 13.1.3 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

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

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

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

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 generated quantities block. alpha $$/$$ beta must be less than $$2 ^ {29}$$. For a description of argument and return types, see section vectorized function signatures.

### References

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