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Discrete distribution describing the number of failures before achieving a fixed number of successes in independent Bernoulli trials.
DESCRIPTION
The negative binomial distribution generalizes the geometric distribution to r successes with success probability p. It models overdispersed count data relative to Poisson, with variance exceeding the mean. Parameterizations include NB1 and NB2 in GLMs, and a gamma–Poisson mixture interpretation. Applications span ecology, epidemiology, and insurance claims. Estimation uses maximum likelihood or Bayesian methods, often with log-link in regression. Zero inflation variants handle excess zeros.
ALTERNATIVE DEFINITIONS
Wolfram MathWorld
The negative binomial distribution gives the probability of x failures before the rth success.
Highlights the classical failure-before-success parameterization.
StatLect
The negative binomial distribution is a discrete distribution that describes the number of failures before a fixed number of successes occurs.
Clear discrete definition with trial interpretation.
Penn State Eberly College of Science
The negative binomial distribution models the number of failures before the rth success in a sequence of independent Bernoulli trials with probability p of success.
Adds independence and Bernoulli trial context.
RELATED TERMS
Binomial Distribution Gamma Distribution Poisson Regression Overdispersion Count Data Generalized Linear Model Zero-Inflated Models Bayesian Inference
COMPETING TERMS
Poisson Distribution Geometric Distribution
RELATIONS
- Broader: Binomial Distribution
KNOWLEDGE AREAS
TARGET AUDIENCE
Data Scientist Data Analyst Data Modeler AI Engineer
EXTERNAL SOURCES
- Wikipedia - Negative binomial distribution
- CRAN - MASS package with NB regression
- UCLA OARC - Applied NB regression tutorial
TOP QUESTIONS
When should I use the Negative Binomial Distribution?
Use it for count data exhibiting overdispersion where Poisson’s equal mean-variance assumption fails.
Category: Relevancy
How does Negative Binomial compare to Poisson?
Negative binomial allows variance > mean, often fitting real-world counts better; Poisson assumes equality.
Category: Alternatives
What are common parameterizations of Negative Binomial?
Parameters include r (size) and p (success probability), or mean–dispersion forms used in GLMs.
Category: Overview
What is a gamma–Poisson mixture interpretation?
Negative binomial can be derived by mixing Poisson rates with a gamma distribution, explaining overdispersion.
Category: Example
What are common mistakes when fitting Negative Binomial regression?
Ignoring zero inflation, misinterpreting dispersion, or using inappropriate links can bias estimates.
Category: Mistakes
How do I check for overdispersion?
Compare residual deviance to degrees of freedom or use dispersion tests; inspect variance relative to mean.
Category: Testing
Can I use Negative Binomial for zero-inflated data?
Yes, zero-inflated NB models combine a structural zero process with NB for counts.
Category: Application
What is the geometric distribution’s relationship to Negative Binomial?
Geometric is a special case with r = 1, measuring failures before the first success.
Category: Overview
How do I interpret NB regression coefficients?
Coefficients on log scale represent multiplicative effects on the expected count; exponentiate to get rate ratios.
Category: Management