Displaced Poisson Distribution

A Poisson-tail count law obtained by conditioning on at least an integer threshold and counting the excess, equivalently shifting the Poisson probability recurrence.

Version

v2 · 2026-08-30 · History

Domain-specific #

1694

Origin domain

probability

Subdomain

discrete distributions

Core Idea

The displaced Poisson distribution is the law of the excess count above a fixed nonnegative integer threshold when an underlying Poisson count is conditioned to reach that threshold. If \(N\sim\operatorname{Poisson}(\lambda)\), \(r\in\mathbb N_0\), and the model conditions on \(N\ge r\), then \(X=N-r\) has support \(x=0,1,2,\ldots\) and

\[ P(X=x)=\frac{e^{-\lambda}\lambda^{x+r}/(x+r)!}{P(N\ge r)}. \]

Staff introduced the construction in precisely this threshold-excess form and emphasized the resulting recurrence and flexibility for dispersed count data. The normalization is not optional: the shifted factorial terms sum only to the Poisson upper-tail probability. The adjacent-probability ratio is.

Scope of Application

The distribution belongs to univariate count modeling and discrete-distribution theory. Its most literal setting is a count observed only after a known threshold has been reached, where the response of interest is the excess beyond that threshold. The construction also supplies a named recursive count family for studying deviations from Poisson equidispersion. Staff's original article described the model as flexible and fitted it to count data; the later Region B study extended the recurrence analysis and parameter estimation.

Clarity

The construction clarifies what “displaced” does. It displaces the factorial index and the observed origin after conditioning, not merely the numerical label on a Poisson variable. Writing the generative form \(X=N-r\mid N\ge r\) makes the normalization, support, and integer constraint visible at once.

Manages Complexity

An infinite probability mass function is compressed into two parameters, one tail-normalizing constant, and a first-order recurrence. Once \(p_0\) is computed, later probabilities require multiplication by \(\lambda/(x+r)\), avoiding repeated factorial evaluation. The conditional interpretation also turns a seemingly bespoke count law into an ordinary Poisson count plus a threshold operation.

Abstract Reasoning

Nesting: setting \(r=0\) is an exact specification check and supports likelihood comparisons with the Poisson baseline. Ratio reasoning: the mode is located where \(\lambda/(x+r)\) crosses one, subject to integer ties. Simulation: draw a Poisson count conditional on clearing \(r\), then subtract \(r\). Normalization: sum the unnormalized weights from \(x=0\) upward and recognize the Poisson tail \(P(N\ge r)\). Interpretation: large \(r\) means the data describe excess beyond a more demanding guaranteed threshold, not a negative location shift.

Knowledge Transfer

Within probability, the construction transfers literally among conditional counting, recursive mass evaluation, distribution fitting, and simulation. The same roles and recurrence persist. The broader lesson—condition on a threshold, recenter at zero, and renormalize—transfers through generic conditional-probability abstractions, but the name displaced Poisson remains tied to a Poisson substrate.

The hyper-Poisson generalization transfers the recurrence shape beyond integer \(r\), but it replaces the threshold story with a special-function normalization. That is a principled extension, not evidence that the specialist vocabulary travels across unrelated domains.

Relationships to Other Abstractions

Current abstraction Displaced Poisson Distribution Domain-specific

Parents (1) — more general patterns this builds on

• Displaced Poisson Distribution is a kind of Probability Distribution Domain-specific

  The displaced Poisson distribution is a strict specialization of domain_specific:probability_distribution: it supplies a normalized law on nonnegative integers with a specific generative claim and readout machinery.

Hierarchy paths (5) — routes to 3 parentless roots

• Displaced Poisson Distribution → Probability Distribution → Random Variable → Function (Mapping)

Show alternative paths (4)

Neighborhood in Abstraction Space

Displaced Poisson Distribution sits in a sparse region of the domain-specific corpus (82^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Variance Gamma Process — 0.83
• Dispersion Function — 0.83
• Doob Decomposition Theorem — 0.83
• Probability Bounds Analysis — 0.81
• Forward–Backward Algorithm — 0.80

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