Hyperbolic distribution

A continuous probability distribution whose log-density traces a hyperbola, yielding exponential but heavier-than-normal tails.

Version

v1 · 2026-09-08 · History

Domain-specific #

4934

Origin domain

probability distribution

Subdomain

probability distribution

Core Idea

It is a specific member of the generalized hyperbolic family rather than any distribution associated with hyperbolic functions, parameterizations differ, and exponential tails are lighter than power-law tails despite being heavier than Gaussian tails. A hyperbolic distance expression in the exponent combines location, scale, asymmetry and shape; normalization produces a density whose logarithm has hyperbolic asymptotes and whose unequal tail rates can model skewness. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application

Hyperbolic distribution belongs to probability distribution and is useful where the analyst can specify the typed probability distribution carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the real-valued random variable and support, location scale shape and asymmetry parameters, density and normalizing constant, hyperbolic log-density geometry, parameter admissibility, exponential tail rates, moments and characteristic function, relation to generalized hyperbolic mixtures, limiting cases and estimation convention are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the real-valued random variable and support, location scale shape and asymmetry parameters, density and normalizing constant, hyperbolic log-density geometry, parameter admissibility, exponential tail rates, moments and characteristic function, relation to generalized hyperbolic mixtures, limiting cases and estimation convention are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Hyperbolic distribution. Hyperbolic distribution compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed probability distribution carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the real-valued random variable and support, location scale shape and asymmetry parameters, density and normalizing constant, hyperbolic log-density geometry, parameter admissibility, exponential tail rates, moments and characteristic function, relation to generalized hyperbolic mixtures, limiting cases and estimation convention are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of probability distribution because they reuse the typed probability distribution carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A hyperbolic distance expression in the exponent combines location, scale, asymmetry and shape; normalization produces a density whose logarithm has hyperbolic asymptotes and whose unequal tail rates can model skewness., and type the carrier, state every parameter and convention in the definition, test that the real-valued random variable and support, location scale shape and asymmetry parameters, density and normalizing constant, hyperbolic log-density geometry, parameter admissibility, exponential tail rates, moments and characteristic function, relation to generalized hyperbolic mixtures, limiting cases and estimation convention are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Hyperbolic distribution Domain-specific

Parents (1) — more general patterns this builds on

• Hyperbolic distribution is a kind of Representation Prime

  The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

• Hyperbolic distribution → Representation → Abstraction

Neighborhood in Abstraction Space

Hyperbolic distribution sits in a moderately populated region (42^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Probability Measures & Random Variables (36 abstractions)

Nearest neighbors

• Hyperbolic growth — 0.90
• Johnson's SU-distribution — 0.90
• Reciprocal distribution — 0.90
• Ratio distribution — 0.90
• Energy distance — 0.89

Computed from structural-signature embeddings · 2026-09-08