Negative Binomial (Pascal)

[LeonidElkin]

Family: Negative Binomial (Pascal)

Naming: NegativeBinomial (negative binomial distribution in the “number of failures before r-th success” parametrization).

Task

Implement the negative binomial family as a ParametricFamily, add all parameterizations and analytical characteristics, and register the family in the registry.

Base parameterization

• Base parameters: (r, p):
  • r > 0 (shape / number of successes, not necessarily integer if using the general form),
  • 0 < p < 1 (success probability).

Interpretation: X is the number of failures before the r-th success.

Support: k in {0, 1, 2, …}.

Other parameterizations and relations

1. Shape–mean (r, μ):
   • Parameters: r > 0, μ > 0 (mean).
   • Relations to base:
     • μ = r * (1 - p) / p
     • p = r / (r + μ).

All characteristics should be defined in terms of (r, p).

Analytical characteristics (in terms of r, p)

For integer k >= 0:

• pmf(k):

  • General (Gamma–function) form:
    pmf(k) = Γ(k + r) / (Γ(r) * k!) * (1 - p)^k * p^r.
    If r is integer, this is C(k + r - 1, k) * (1 - p)^k * p^r.

• cdf(k):

  • cdf(k) = sum_{j=0}^k pmf(j).
    It can also be expressed via the regularized incomplete beta function.

• ppf(u) for 0 < u < 1:

  • ppf(u) is the smallest integer k >= 0 such that cdf(k) >= u.

• cf(t):

  • cf(t) = (p / (1 - (1 - p) * exp(i * t)))^r.

• mean:

  • mean = r * (1 - p) / p.

• var:

  • var = r * (1 - p) / p^2.