Negative binomial distribution

A discrete probability distribution for the number of failures before a specified number of successes in independent Bernoulli trials with constant success probability.

Bernoulli trial: A random experiment with two outcomes, usually called success and failure, occurring with a fixed probability. Each independent trial supplies one success or failure in the sequence being counted.
Probability: A numerical measure of how likely an event is, ranging from zero to one. Success probability sets the chance of every Bernoulli outcome in the model.
Negative binomial regression: A regression model for count outcomes that allows variance to exceed the conditional mean. It uses this distribution to model counts whose variability exceeds a Poisson model's.
Binomial distribution: The distribution of the number of successes in a fixed number of independent Bernoulli trials. It fixes the trial count, while the negative binomial fixes the required success count.
Negative binomial probability mass function: The formula assigning probabilities to possible failure counts in a negative binomial distribution. It combines the final success, the earlier successes, and all possible failure placements.
Independence (probability theory): A relationship in which knowing one event occurred does not change the probability of another. The trial-sequence formula assumes each outcome leaves later trial probabilities unchanged.
Overdispersion: Greater variability in observed data than a statistical model predicts. The negative binomial model accommodates count data with variance above its mean.
Pascal distribution: A discrete distribution for the number of trials needed to obtain a specified number of successes. It counts total trials rather than failures, differing by the fixed number of successes.
Binomial coefficient: The number of ways to choose a fixed-size subset from a larger set. It counts the ways failures can appear before the final required success.
Expected value: The probability-weighted average value of a random variable. It gives the distribution's average number of failures before the target is reached.

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