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.^[1] The normalization is not optional: the shifted factorial terms sum only to the Poisson upper-tail probability. The adjacent-probability ratio is

\[ \frac{p_x}{p_{x-1}}=\frac{\lambda}{x+r},\qquad x\ge1, \]

which is the Poisson ratio \(\lambda/x\) with a displaced denominator. At \(r=0\), the conditioning event is certain, \(X=N\), and the ordinary Poisson law is recovered.

This strict identity matters because “displaced Poisson” and “hyper-Poisson” are sometimes used loosely as synonyms. The authoritative distribution reference by Johnson, Kemp, and Kotz treats integer displacement as a special case of the broader hyper-Poisson family, whose factorial is generalized through gamma or hypergeometric normalization.^[2] This entry preserves the classical integer-threshold construction; real-parameter hyper-Poisson and Staff's separately analyzed nonpositive-parameter region are related extensions, not silently identical definitions.^[3]

Structural Signature

Recognition roles: an underlying Poisson count \(N\) — a rate \(\lambda>0\) — a nonnegative integer threshold \(r\) — conditioning on \(N\ge r\) — an excess count \(X=N-r\) — upper-tail normalization — the displaced recurrence \(\lambda/(x+r)\) — Poisson recovery at \(r=0\).

• Poisson substrate. The unconditioned count has mass \(e^{-\lambda}\lambda^n/n!\).
• Threshold and condition. The event \(N\ge r\) is imposed before displacement; subtracting \(r\) without conditioning would retain negative outcomes.
• Excess support. The modeled variable begins at zero and records events beyond the guaranteed threshold.
• Renormalized shifted factorial. The denominator \((x+r)!\) and tail normalization jointly define the law.
• Recurrence. Successive masses have ratio \(\lambda/(x+r)\), permitting stable computation from \(p_0\).
• Nested baseline. \(r=0\) yields the ordinary Poisson distribution exactly.
• Taxonomic boundary. Continuous displacement parameters require the hyper-Poisson extension and must not be smuggled into the threshold interpretation.

Recognition test: demand either the conditional construction \(N-r\mid N\ge r\) or its equivalent normalized mass and recurrence with integer \(r\ge0\). A count law merely shifted by adding a constant, or any flexible two-parameter Poisson generalization, fails.

What It Is Not

• It is not a simple location shift \(N+c\). A location shift changes support but preserves the original probability ratios after reindexing; displaced Poisson also conditions and renormalizes a tail.
• It is not the ordinary Poisson distribution except at \(r=0\). The denominator in the recurrence and the mass at zero differ.
• It is not automatically the entire hyper-Poisson family. Hyper-Poisson permits a noninteger shape parameter and uses confluent-hypergeometric normalization; integer displaced Poisson is a nested case.^[2]
• It is not the generalized Poisson or Conway–Maxwell–Poisson distribution. Those modify probability recurrences differently and carry different parameter interpretations.
• It is not a Poisson process. It is a one-variable probability law; a process requires counts indexed by time or another exposure with increment structure.
• It is not truncation without recentering. A zero-truncated Poisson \(N\mid N\ge1\) has support \(1,2,\ldots\); the \(r=1\) displaced law records \(X=N-1\) on \(0,1,\ldots\).

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.^[1][3]

Modern count regression more often uses the broader hyper-Poisson family, which supports under- and overdispersion through a continuously varying parameter. Sáez-Castillo and Conde-Sánchez develop that regression model and explicitly position it against the equal-mean-and-variance restriction of Poisson regression.^[4] This is relevant lineage but not permission to assign every hyper-Poisson regression to the integer-threshold distribution.

Applications qualify as displaced Poisson only when the fitted mass function or recurrence matches the locked identity. Merely citing thresholded sampling, excess counts, or overdispersion is insufficient. In particular, hurdle, zero-inflated, and zero-truncated models have different mass reallocation rules.

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.

The recurrence supplies a second diagnostic. If fitted adjacent masses satisfy \(p_x/p_{x-1}=\lambda/(x+r)\), the denominator intercept \(r\) controls how quickly the probabilities decline. But recurrence alone must be paired with declared support: for negative or noninteger parameters, positivity and starting index become separate questions. This is why the strict node avoids presenting one formula across all real \(r\) as if no boundary changed.

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.

The abstraction keeps the rate, threshold, support, and normalization explicit. It discards the timing of individual events and any mechanism that makes increments dependent. In regression use, covariate links and observation-specific dispersion are additional modeling layers, not part of the base distribution.

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.

Moments should be computed from the conditional Poisson tail rather than copied from the unconditioned law. In particular, the seed claim “mean \(=\lambda-r\), variance \(=\lambda\)” is not generally valid after tail conditioning for positive \(r\). The conditional event changes moments. This correction is load-bearing: retaining ordinary Poisson cumulants would contradict the very renormalization that defines the distribution.

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.

Examples

One-event threshold

Let \(N\sim\operatorname{Poisson}(2)\), condition on \(N\ge1\), and set \(X=N-1\). Then

\[ p_x=\frac{e^{-2}2^{x+1}/(x+1)!}{1-e^{-2}},\qquad x\ge0. \]

The first ratio is \(p_1/p_0=2/(1+1)=1\); the next is \(p_2/p_1=2/3\). Thus zero and one excess event tie at the top before the tail declines. The example maps the Poisson substrate, threshold, condition, excess support, normalization, and displaced recurrence.

Poisson boundary

Set \(r=0\). Then \(P(N\ge0)=1\), \(X=N\), and

\[ p_x=e^{-\lambda}\frac{\lambda^x}{x!},\qquad \frac{p_x}{p_{x-1}}=\frac{\lambda}{x}. \]

Every displaced role collapses to the ordinary Poisson law. This is both a worked example and a unit test for implementations: failure to recover Poisson at zero displacement signals an indexing or normalization error.

Structural Tensions

T1: Generative threshold meaning versus algebraic parameter extension. Integer \(r\) has a literal event-threshold interpretation; real hyper-Poisson parameters improve flexibility but lose that story. Diagnostic: Can the fitted parameter be represented as a count threshold, or is it only a shape parameter in a gamma-normalized family?

T2: Flexible dispersion versus model interpretability. Modifying the recurrence can fit counts that violate Poisson equidispersion, but a good fit does not prove a threshold-conditioned Poisson mechanism. Diagnostic: Is the model being used descriptively, or does the sampling process independently support the condition-and-excess construction?

T3: Simple recurrence versus support fragility. The ratio is computationally compact, yet nonpositive or noninteger displacement can make denominators, positivity, and starting indices delicate. Diagnostic: Have support and parameter region been declared before iterating the recurrence?

T4: Autonomous law versus Probability Distribution plus Conditioning. Generic nodes describe the ingredients but not the shifted factorial, exact recurrence, nested Poisson boundary, and threshold-excess interpretation. Diagnostic: Can the composite distinguish this law from zero-truncated, hurdle, and hyper-Poisson variants without restating the named mass function?

Structural–Framed Character

This is a structurally framed mathematical abstraction. Equations determine its validity, while statistical practice determines whether it is a useful model. Its vocabulary—probability mass, Poisson rate, recurrence, conditional tail, normalization—is specialist and does not travel unchanged outside count theory.

Terminological practice does vary: some literature uses hyper-Poisson broadly and displaced Poisson for a subset, while older papers analyze multiple parameter regions under the displaced name. The node handles that framing by locking a strict core and recording extensions explicitly.

Structural Core vs. Domain Accent

The portable core is condition–recenter–renormalize plus recursive generation of weights. The domain accent is an integer Poisson count, factorial mass, upper-tail probability, and Poisson nesting. Removing those yields a generic truncated distribution.

The candidate does not clear the prime bar: it recurs within count statistics, not across independent substrates. It clears the domain-specific bar through a stable construction, recognition test, formula, diagnostics, and distinct modeling consequences.

Instantiates / Related Primes

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. That is the proposed minimal parent.

It also uses prime:conditional_probability, but conditioning is an ingredient rather than the taxonomic genus. prime:poisson_process is declined because a distribution of one count does not imply an indexed process with independent increments. domain_specific:delaporte_distribution is a sibling count family with a different compound construction.

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

Not to Be Confused With

• Hyper-Poisson distribution: broader real-parameter gamma/hypergeometric family; displaced Poisson is the integer-threshold case under the strict taxonomy used here.
• Zero-truncated Poisson: conditions away zero but keeps support starting at one; the displaced version recenters the threshold at zero.
• Shifted Poisson: adds a constant without conditioning or tail renormalization.
• Generalized Poisson distribution: modifies the mass function through a different mean–dispersion parameterization.
• Conway–Maxwell–Poisson: changes factorial powers to control dispersion.
• Delaporte distribution: convolution of a Poisson and negative-binomial component, not a threshold excess.

References

[1] P. J. Staff, “The Displaced Poisson Distribution,” Australian Journal of Statistics 6, no. 1 (1964): 12–20, https://doi.org/10.1111/j.1467-842X.1964.tb00146.x. registry ↩a ↩b

[2] Norman L. Johnson, Adrienne W. Kemp, and Samuel Kotz, Univariate Discrete Distributions, 3^rd ed. (Wiley, 2005), https://doi.org/10.1002/0471715816. registry ↩a ↩b

[3] P. J. Staff, “The Displaced Poisson Distribution—Region B,” Journal of the American Statistical Association 62, no. 318 (1967): 643–654, https://doi.org/10.1080/01621459.1967.10482938. registry ↩a ↩b

[4] Antonio J. Sáez-Castillo and Antonio Conde-Sánchez, “A Hyper-Poisson Regression Model for Overdispersed and Underdispersed Count Data,” Computational Statistics & Data Analysis 61 (2013): 148–157, https://doi.org/10.1016/j.csda.2012.12.009. registry ↩