Family: Poisson

[LeonidElkin]

Family: Poisson

Naming: Poisson (Poisson distribution).

Task

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

Base parameterization

• Base parameter: λ > 0 (intensity / rate).

Support: k in {0, 1, 2, …}.

Other parameterizations and relations

1. Log-rate (η):
   • Parameter: η in ℝ.
   • Relations to base:
     • λ = exp(η)
     • η = log(λ).

All characteristics should be defined in terms of λ.

Analytical characteristics (in terms of λ)

• pmf(k) for integer k >= 0:

  • pmf(k) = exp(-λ) * λ^k / k!.

• cdf(k):

  • cdf(k) = sum_{j=0}^k pmf(j).

• ppf(u) for 0 < u < 1:

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

• cf(t):

  • cf(t) = exp(λ * (exp(i * t) - 1)).

• mean:

  • mean = λ.

• var:

  • var = λ.