12.10 Negative Binomial Distribution

• Like the Poisson model, the Negative Binomial is suitable for count data \(Y\in\{0,1,2,…\}\)
• Unlike the Poisson, the Negative Binomial does not make the restrictive assumption that \(\text{E}(Y)=\text{Var}(Y)\)
• \(\mu\): mean parameter
• \(r\): reciprocal dispersion parameter

\[\begin{array}{rcl} Y|\mu, r & \sim & \text{NegBin}(\mu,r) \\ f(y|\mu,r) & = & \binom{y+r-1}{r}\left(\frac{r}{\mu+r}\right)^{r}\left(\frac{\mu}{\mu+r}\right)^{y} \\ \text{E}(Y|\mu, r) & = & \mu \\ \text{Var}(Y|\mu, r) & = & \mu + \frac{\mu^{2}}{r} \\ \end{array}\]