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Here the negative binomial distribution has mean parameter $\mu >0$ and size parameter $\theta >0$. Its support in the original PDF is all nonnegative integers; a finite endpoint appearing in the TeX aid would be incorrect. The probability generating function, obtained from the generalized binomial series, isDifferentiating at one gives $G^{\prime }(1)=\mu$ and $G^{\prime \prime }(1)={\mu }^2(1+1/\theta )$. Since the variance is $G^{\prime \prime }(1)+G^{\prime }(1)-G^{\prime }(1{)}^2$, this provesThe extra positive term explains its overdispersion relative to a Poisson distribution.

For one observation, write the log-likelihood asThe mean score function and its cross derivative areTaking expectations gives $I_{\mu \theta }=-\mathbb{E}({\partial }_{\theta }{u}_{\mu })=0$. In addition,For reference, with $\psi$ the digamma function, the size score function isIts variance $I_{\theta \theta }$ is finite and positive for finite positive $\mu ,\theta$. Thus negative binomial mean-size parameter orthogonality makes the expected Fisher information matrix diagonal. For an independent identically distributed sample and a regular identifiable interior true parameter, asymptotic normality of a maximum likelihood estimator givesHence the asymptotic correlation is zero. This is first-order asymptotic independence, not a claim of exact finite-sample independence. Indeed the mean score yields $\hat{\mu }=\bar{y}$ at an interior fit, while the size fit also depends on the observed dispersion. The argument excludes a true boundary such as the Poisson limit $\theta =\infty$.