Which distribution is commonly used for count data with overdispersion, extending the Poisson model?

• A. Normal
• B. Negative binomial
• C. Binomial
• D. Gamma

Answer: (B)

Count data are often modeled with Poisson, which assumes the mean equals the variance. When you observe overdispersion—variance larger than the mean—the Negative Binomial fits better because it adds extra variability. Conceptually, if the Poisson rate varies across observations according to a Gamma distribution, the resulting marginal distribution of counts becomes Negative Binomial. In regression form, you model the mean count as mu_i = exp(X_i beta), and include a dispersion parameter that inflates the variance to mu_i + phi mu_i^2, capturing overdispersion. If phi were zero, you’d revert to Poisson.

Other options don’t fit the count-with-overdispersion situation: Normal is for continuous data and isn’t discrete-count appropriate; Binomial applies to counts of successes in a fixed number of trials with a fixed probability; Gamma is for continuous positive data, not integers.