Negative Binomial Distribution
X ~ NegBinom(r, p)
Overdispersed Count Family • Published July 19, 2026
The standard fix for overdispersed count data: history from Greenwood and Yule's 1920 accident-proneness study through the exact Gamma-Poisson mixture representation, why variance always exceeds the mean (and only approaches Poisson's equidispersion in a limit), maximum-likelihood estimation of the success probability, and simulation via a sum of Geometric draws.
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About This Reference Sheet
The standard fix for overdispersed count data: history from Greenwood and Yule's 1920 accident-proneness study through the exact Gamma-Poisson mixture representation, why variance always exceeds the mean (and only approaches Poisson's equidispersion in a limit), maximum-likelihood estimation of the success probability, and simulation via a sum of Geometric draws.
Support
x ∈ {0, 1, 2, ...}
Parameters
r > 0 (target successes), p ∈ (0, 1] (success probability)