Family: Continuous Uniform

[LeonidElkin]

Family: Continuous Uniform

Naming: Uniform or ContinuousUniform (continuous uniform distribution).

Task

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

Base parameterization

• Base parameters: (a, b) with constraint a < b.

All other parameterizations must be expressed in terms of (a, b).

Other parameterizations and relations

1. Center–radius parameterization (m, r):

   • Parameters:
     • m – center (real),
     • r > 0 – half-length.
   • Relation to base:
     • a = m - r
     • b = m + r

2. Special case [0, θ] parameterization (θ):

   • Parameter: θ > 0.
   • Relation to base:
     • a = 0
     • b = θ

Any characteristic evaluation in alternative parameterizations should internally convert to (a, b).

Analytical characteristics (in terms of a, b)

Let n = b - a > 0.

• pdf(x):

  • pdf(x) = 1 / n if a <= x <= b,
  • pdf(x) = 0 otherwise.

• cdf(x):

  • cdf(x) = 0 if x < a,
  • cdf(x) = (x - a) / n if a <= x <= b,
  • cdf(x) = 1 if x > b.

• ppf(u) (0 < u < 1):

  • ppf(u) = a + u * n.

• cf(t) (characteristic function):

  • for t != 0: cf(t) = exp(i * t * a) * (exp(i * t * n) - 1) / (i * t * n),
  • for t = 0: cf(0) = 1.

• mean:

  • mean = (a + b) / 2.

• var:

  • var = n^2 / 12.