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.

Version: v2 · 2026-09-19 · History
Domain-specific #: 1636
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability and Statistics, Overdispersed Count Distributions → Mathematics
Aliases: Poisson–negative-binomial convolution distribution

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.

How would you explain it like I'm…

Steady Cloud, Moody Cloud

Imagine counting raindrops in a bucket from two clouds. One cloud rains at a steady speed. The other cloud is moody: some days it rains a lot and some days a little. The total count from both clouds follows the Delaporte Distribution.

Steady Count Plus Wobbly Count

Some things we count happen at random, like how many emails arrive in an hour. A Poisson count is what you get when the average rate is steady. A negative binomial count is what you get when the rate itself goes up and down randomly. The Delaporte Distribution is what you get when you add one count of each kind together: part of the rate is fixed and part is random. It is more spread out than a plain steady count, but usually less spread out than a fully random-rate count.

Poisson Plus Negative Binomial Counts

The Delaporte Distribution is a probability distribution for counts (0, 1, 2, …). It arises by adding two independent counts: X, which is Poisson with mean λ, and Y, which is negative binomial. Equivalently, you can think of a single Poisson count whose rate is λ + G, where λ is a fixed part and G is a gamma-distributed random part. The random part makes the counts more variable than a Poisson with the same average, so the distribution sits between Poisson and negative binomial in how dispersed it is. Different textbooks write the parameters differently, so you must say which convention you are using.

The Delaporte Distribution is the distribution of N = X + Y, where X ~ Poisson(λ) and Y is an independent negative binomial variable with a declared shape and scale parameterization. Equivalently, it is a mixed Poisson distribution: conditional on a random intensity Λ = λ + G, where G is gamma-distributed, N ~ Poisson(Λ). The fixed component λ gives a baseline of Poisson variation, while the gamma component adds extra heterogeneity, so the distribution occupies an intermediate dispersion regime between pure Poisson (no random intensity) and negative binomial (entirely gamma-random intensity). Because sources parameterize the negative binomial and gamma parts differently, any formula must declare whether its second parameter is a scale, rate, probability or mean component. The defining structure is exactly this Poisson-plus-negative-binomial decomposition of a count.

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) — outgoing typed relations

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 (3) — routes to 2 parentless roots

Delaporte Distribution → is a kind of Probability Distribution → presupposes, typical Random Variable → is a kind of Function (Mapping)

Show alternative paths (2)

Neighborhood in Abstraction Space¶

Delaporte Distribution sits in a sparse region of the domain-specific corpus (79^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Composite Probability Distributions (5 abstractions)

Nearest neighbors

Yule–Simon Distribution — 0.83
Covariance Matrix — 0.82
Zero-truncated Poisson distribution — 0.82
Benford's Law — 0.82
Exponentially Modified Gaussian Distribution — 0.82

Computed from structural-signature embeddings · 2026-10-25