Family: Bernoulli

[LeonidElkin]

Family: Bernoulli

Naming: Bernoulli (discrete distribution on {0, 1}).

Task

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

Base parameterization

• Base parameter: p with 0 < p < 1.

Support: k in {0, 1}.

Other parameterizations and relations

1. Odds (ω):

   • Parameter: ω > 0.
   • Relations to base:
     • p = ω / (1 + ω)
     • ω = p / (1 - p).

2. Logit (η):

   • Parameter: η in ℝ.
   • Relations to base:
     • p = 1 / (1 + exp(-η))
     • η = log(p / (1 - p)).

All characteristics should be defined in terms of p.

Analytical characteristics (in terms of p)

• pmf(k) for k in {0, 1}:
  • pmf(1) = p,
  • pmf(0) = 1 - p.

A compact form: pmf(k) = p^k * (1 - p)^(1 - k).

• cdf(k):

  • cdf(0) = P(X <= 0) = 1 - p,
  • cdf(1) = 1.

• ppf(u) (0 <= u <= 1):

  • ppf(u) = 0 if u <= 1 - p,
  • ppf(u) = 1 if u > 1 - p.

• cf(t):

  • cf(t) = (1 - p) + p * exp(i * t).

• mean:

  • mean = p.

• var:

  • var = p * (1 - p).