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## 13.2 Negative Binomial Distribution (alternative parameterization)

Stan also provides an alternative parameterization of the negative binomial distribution directly using a mean (i.e., location) parameter and a parameter that controls overdispersion relative to the square of the mean. Section combinatorial functions, below, provides a second alternative parameterization directly in terms of the log mean.

### 13.2.1 Probability Mass Function

The first parameterization is for $$\mu \in \mathbb{R}^+$$ and $$\phi \in \mathbb{R}^+$$, which for $$n \in \mathbb{N}$$ is defined as $\text{NegBinomial2}(n \, | \, \mu, \phi) = \binom{n + \phi - 1}{y} \, \left( \frac{\mu}{\mu+\phi} \right)^{\!y} \, \left( \frac{\phi}{\mu+\phi} \right)^{\!\phi} \!.$

The mean and variance of a random variable $$n \sim \text{NegBinomial2}(n~|~\mu,\phi)$$ are $\mathbb{E}[n] = \mu \ \ \ \text{ and } \ \ \ \text{Var}[n] = \mu + \frac{\mu^2}{\phi}.$ Recall that $$\text{Poisson}(\mu)$$ has variance $$\mu$$, so $$\mu^2 / \phi > 0$$ is the additional variance of the negative binomial above that of the Poisson with mean $$\mu$$. So the inverse of parameter $$\phi$$ controls the overdispersion, scaled by the square of the mean, $$\mu^2$$.

### 13.2.2 Sampling Statement

n ~ neg_binomial_2(mu, phi)

Increment target log probability density with neg_binomial_2_lpmf(n | mu, phi) dropping constant additive terms.

### 13.2.3 Stan Functions

real neg_binomial_2_lpmf(ints n | reals mu, reals phi)
The negative binomial probability mass of n given location mu and precision phi.

real neg_binomial_2_cdf(ints n, reals mu, reals phi)
The negative binomial cumulative distribution function of n given location mu and precision phi.

real neg_binomial_2_lcdf(ints n | reals mu, reals phi)
The log of the negative binomial cumulative distribution function of n given location mu and precision phi.

real neg_binomial_2_lccdf(ints n | reals mu, reals phi)
The log of the negative binomial complementary cumulative distribution function of n given location mu and precision phi.

R neg_binomial_2_rng(reals mu, reals phi)
Generate a negative binomial variate with location mu and precision phi; may only be used in transformed data and generated quantities blocks. mu must be less than $$2 ^ {29}$$. For a description of argument and return types, see section vectorized function signatures.