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: v3 · 2026-09-19 · History
Domain-specific #: 1694
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability, Discrete Distributions → Mathematics

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.

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Counting the Extras

Suppose you count shooting stars each night, but you only keep the nights when you saw at least three. Then, for those nights, you write down how many extra you saw beyond three. The pattern of how often you get each number of 'extras' is the displaced Poisson distribution.

Extra Counts Past a Starting Line

The Poisson distribution describes counts of things that happen randomly, like how many shooting stars you see in a night. The displaced Poisson distribution is what you get when you only look at counts that reach some fixed number, say 3, and then record how far above 3 each count is. So a count of 3 becomes 0, a count of 5 becomes 2, and so on. Because you threw away the counts below 3, you have to rescale the chances so they still add up to 1. If the threshold is 0, nothing gets thrown away and you just get the ordinary Poisson distribution back.

Threshold-Excess Poisson Law

Start with a Poisson count N with rate λ, and a fixed whole number r ≥ 0. The displaced Poisson distribution is the distribution of X = N - r, given that N ≥ r: the excess above the threshold, with possible values 0, 1, 2, and so on. Its probabilities are the Poisson probabilities for N = x + r divided by P(N ≥ r); that division is essential, because the shifted terms only add up to the upper-tail probability. The ratio of neighboring probabilities is λ/(x + r), compared to λ/x for ordinary Poisson, and this recurrence gives the model extra flexibility for count data whose spread doesn't match a plain Poisson. Setting r = 0 gives back the ordinary Poisson distribution. The name is sometimes used loosely for the broader 'hyper-Poisson' family, but this is the specific integer-threshold version.

The displaced Poisson distribution is the law of the excess over a fixed nonnegative integer threshold r of a Poisson(λ) count N, conditioned on N ≥ r. With X = N - r and support x = 0, 1, 2, ..., its probability mass function is P(X = x) = [e^{-λ} λ^{x+r} / (x+r)!] / P(N ≥ r). The normalization by the Poisson upper-tail probability is required, since the shifted terms alone sum to that tail rather than to one. Adjacent probabilities satisfy p_x / p_{x-1} = λ / (x + r) for x ≥ 1, a Poisson-type recurrence with a displaced denominator, which Staff emphasized as a source of flexibility for dispersed count data. When r = 0 the conditioning event is certain and the ordinary Poisson law is recovered. The term is sometimes used interchangeably with 'hyper-Poisson', but Johnson, Kemp, and Kotz treat integer displacement as a special case of the broader hyper-Poisson family, whose factorial is generalized via gamma or hypergeometric normalization; real-parameter versions and Staff's nonpositive-parameter region are related extensions, not the same definition.

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) — outgoing typed relations

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 (3) — routes to 2 parentless roots

Displaced Poisson 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¶

Displaced Poisson Distribution sits in a sparse region of the domain-specific corpus (84^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

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