Family: Binomial

[LeonidElkin]

Family: Binomial

Naming: Binomial (binomial distribution).

Task

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

Base parameterization

• Base parameters: (n, p):
  • n in ℕ (number of trials),
  • 0 < p < 1 (success probability).

Support: k in {0, 1, …, n}.

Other parameterizations and relations

1. Logit parameterization (n, η):

   • Parameters: n in ℕ, η in ℝ.
   • Relations to base:
     • p = 1 / (1 + exp(-η))
     • η = log(p / (1 - p)).

2. Mean parameterization (n, μ):

   • Parameters: n in ℕ, μ such that 0 < μ < n.
   • Relations to base:
     • p = μ / n
     • μ = n * p.

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

Analytical characteristics (in terms of n, p)

• pmf(k) for integer 0 <= k <= n:

  • pmf(k) = C(n, k) * p^k * (1 - p)^(n - k),
    where C(n, k) is the binomial coefficient.

• cdf(k):

  • cdf(k) = sum_{j=0}^k pmf(j).

• ppf(u) for 0 < u < 1:

  • ppf(u) is the smallest integer k in {0, …, n} such that cdf(k) >= u.

• cf(t):

  • cf(t) = (1 - p + p * exp(i * t))^n.

• mean:

  • mean = n * p.

• var:

  • var = n * p * (1 - p).