Delaporte Distribution¶
A discrete count distribution formed as the convolution of independent Poisson and negative-binomial components, equivalently a Poisson count with a mean containing fixed and gamma-random parts.
Core Idea¶
Delaporte Distribution is a discrete count distribution formed as the convolution of independent Poisson and negative-binomial components, equivalently a Poisson count with a mean containing fixed and gamma-random parts.
Let X be Poisson with mean lambda and Y be independent negative binomial with a declared shape/scale parameterization. Then N=X+Y has a Delaporte distribution. Equivalently, conditional on a random intensity Lambda=lambda+G with G gamma-distributed, N is Poisson(Lambda). Parameterizations differ across sources, so formulas must declare whether beta is a scale, rate, probability, or mean component.
Scope of Application¶
The abstraction has a bounded but recurring habitat. These are literal applications of the same domain machinery, not cross-domain metaphors.
- Actuarial claim counts. baseline events and heterogeneous risk can be represented in one frequency law.
- Epidemic offspring counts. fixed and variable transmissibility components create intermediate dispersion.
- Ecological abundance. background counts and clustered variation can coexist.
- Genomic count models. signal plus overdispersed biological component motivates convolution likelihoods.
- Queueing and reliability. independent baseline and environment-driven counts can yield the same law.
- Simulation studies. the additive construction gives a direct random-generation algorithm.
Clarity¶
The safest specification is constructive: name independent X and Y and their exact parameterizations, then set N=X+Y. Mean and variance follow by addition. Under a common gamma scale beta and shape alpha, E[N]=lambda+alpha beta and Var[N]=lambda+alpha beta(1+beta).
Manages Complexity¶
The family separates a constant Poisson contribution from heterogeneous overdispersion while retaining a closed count model. It interpolates between Poisson and negative binomial behavior and supplies a simulation story, probability-generating function, and moment structure from component laws.
The compression remains accountable because every simplification has a named validity condition. A user can ask which role is missing, which assumption fails, and which neighboring abstraction should replace the candidate instead of treating the label as an unanalyzed bundle.
Abstract Reasoning¶
R1. Lock the negative-binomial parameterization before copying a mass function.
R2. Use convolution only when the components are independent.
R3. Check limiting cases by setting each component to zero in turn.
R4. Compare mean and variance to identify, not prove, overdispersion structure.
R5. Distinguish parameter identifiability from numerical optimizer convergence.
Knowledge Transfer¶
The law transfers literally wherever nonnegative counts plausibly decompose into independent Poisson and negative-binomial components. The broader convolution abstraction travels to other variables; the Delaporte name stays with this exact pair and parameter family.
The transfer boundary follows from the classification test: The family recurs in actuarial, epidemiological, and count-data models, but its three parameters, independence, convolution, support, overdispersion envelope, and limiting cases are constitutive. The safe portable move is to name the broader parent when the home-domain machinery is absent and to retain the domain name only when literal recognition succeeds.
Relationships to Other Abstractions¶
Current abstraction Delaporte Distribution Domain-specific
Parents (1) — more general patterns this builds on
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Delaporte Distribution is a kind of Probability Distribution Domain-specific
The accepted reference-grade review places Delaporte Distribution under Probability Distribution because the child instantiates or depends on the parent's broader structure while retaining its own constitutive identity.
Hierarchy paths (5) — routes to 3 parentless roots
- Delaporte Distribution → Probability Distribution → Random Variable → Function (Mapping)
- Delaporte Distribution → Probability Distribution → Probability → Measure → Set and Membership
- Delaporte Distribution → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Delaporte Distribution → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Delaporte Distribution → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Delaporte Distribution sits in a sparse region of the domain-specific corpus (81st percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Yule–Simon Distribution — 0.83
- Benford's Law — 0.82
- Polykay — 0.82
- Variance function — 0.82
- Empirical Measure — 0.81
Computed from structural-signature embeddings · 2026-09-08