## 17.2 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.

### 17.2.1 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 $\begin{equation*} \text{NegBinomial2}(n \, | \, \mu, \phi) = \binom{n + \phi - 1}{n} \, \left( \frac{\mu}{\mu+\phi} \right)^{\!n} \, \left( \frac{\phi}{\mu+\phi} \right)^{\!\phi} \!. \end{equation*}$

The mean and variance of a random variable $$n \sim \text{NegBinomial2}(n~|~\mu,\phi)$$ are $\begin{equation*} \mathbb{E}[n] = \mu \ \ \ \text{ and } \ \ \ \text{Var}[n] = \mu + \frac{\mu^2}{\phi}. \end{equation*}$ 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, $$\mu^2$$.

### 17.2.2 Sampling statement

n ~ neg_binomial_2(mu, phi)

Increment target log probability density with neg_binomial_2_lupmf(n | mu, phi).
Available since 2.3

### 17.2.3 Stan functions

real neg_binomial_2_lpmf(ints n | reals mu, reals phi)
The log negative binomial probability mass of n given location mu and precision phi.
Available since 2.20

real neg_binomial_2_lupmf(ints n | reals mu, reals phi)
The log negative binomial probability mass of n given location mu and precision phi dropping constant additive terms.
Available since 2.25

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.
Available since 2.6

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.
Available since 2.12

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.
Available since 2.12

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.
Available since 2.18