Negative Binomial Distribution

(discrete probability dist. for equations)

 

Usage:

NegBinomialDist (k, n, p)

Definition:

binomial (n+k-1, k) p^n (1-p)^k

where binomial is the binomial coefficient.

Required:

0 ⋜ n   0 < p ⋜ 1   k and n are integers

 

Support:

0 ⋜ k

Moments:

μ = n (1-p) / p

σ^2 = n (1-p) / p^2

γ1 = (2 - p) / sqrt (n (1 - p))

β2 = 3 + [p^2 + 6 (1-p)] / (n (1 - p))

 

This is the distribution of the number of failures that occur in a sequence of trials before n successes have occurred, in a Bernoulli process (independent trials, with outcomes labeled "success" or "failure", and constant probability p of success).

The limit of a negative binomial distribution as n → ∞, (1-p) → 0, n(1-p) → λ, is a Poisson distribution with parameter λ.

If n = 1, then this distribution is just the geometric distribution.