Poisson binomial distribution¶
Describe the success count obtained by summing independent Bernoulli variables whose success probabilities may differ, retaining the full heterogeneous probability vector.
Core Idea¶
The Poisson binomial distribution is the law of \(S=\sum_{i=1}^{n}X_i\) when \(X_i\) are independent Bernoulli variables with success probabilities \(p_i\) that need not be equal. Its support is \(\{0,1,\ldots,n\}\). The probability of \(k\) successes sums \(\prod_{i\in A}p_i\prod_{j\notin A}(1-p_j)\) over all size-\(k\) subsets \(A\). Equal probabilities reduce the law to the ordinary binomial distribution, but heterogeneity is retained in the general case.
Independence makes the probability-generating function factor as \(G(z)=\prod_{i=1}^{n}(1-p_i+p_i z)\); the coefficient of \(z^k\) is \(\Pr(S=k)\). The mean and variance are \(\sum_i p_i\) and \(\sum_i p_i(1-p_i)\).
Scope of Application¶
The abstraction is literal wherever practitioners can identify the same constitutive roles, apply the same boundary tests, and obtain the same kind of output. The following habitats are uses of Poisson binomial distribution itself, not metaphors based only on resemblance.
- Reliability. Counting functioning components with unequal independent reliabilities.
- Survey and auditing. Modeling heterogeneous independent inclusion or error indicators.
- Risk aggregation. Counting occurrences across exposures with unequal probabilities under an independence model.
- Voting models. Counting independent approvals with voter-specific probabilities.
- Exact testing. Computing a heterogeneous-null count distribution without replacing probabilities by their mean.
- Approximation analysis. Quantifying when binomial, normal, or Poisson substitutions are adequate.
Clarity¶
A clear account of Poisson binomial distribution must preserve the recognition invariant stated in the Core Idea rather than rely on the title alone. List the probability vector and justify independence at the modeled level. State whether the desired object is the count law, a conditioned configuration law, or an approximation. Handle probabilities equal to zero or one explicitly before using odds recurrences. Validate numerical probabilities for nonnegativity, unit sum, and agreement with analytic moments.
Manages Complexity¶
Poisson binomial distribution manages complexity by replacing a diffuse field of observations or possible operations with a bounded role structure: trial index supplies a finite collection identifies the individual Bernoulli components.; binary outcomes supplies each component contributes zero or one to the count.; heterogeneous probabilities supplies each trial carries its own parameter between zero and one.; independence supplies the joint law factors across components and enables the generating product.; success count supplies the sum forgets which trials succeeded and retains only how many..
Abstract Reasoning¶
- Define one Bernoulli indicator per trial and record its success probability. 2. Assess whether conditional or unconditional independence is the defensible assumption. 3. Form the generating polynomial or initialize an equivalent convolution recurrence. 4. Extract coefficients for the required probabilities while tracking numerical error. 5. Check that masses sum to one and reproduce the theoretical mean and variance. 6. Compare with the equal-probability binomial special case and any proposed approximation.
Knowledge Transfer¶
The strict upward abstraction is Probability. Poisson Binomial Distribution instantiates Probability because it assigns normalized likelihoods to every possible success count under an explicitly stated random model. Within heterogeneous bernoulli sums, the full mechanism transfers literally when the same roles and boundary tests recur. Beyond that domain, only the parent-level skeleton should travel. Reusing the label Poisson binomial distribution after removing its constitutive vocabulary would hide a change of mechanism behind an analogy. The honest transfer rule is therefore two-stage: recognize the domain-specific pattern first, then lift only the parent relation that remains invariant under a substrate change.
Relationships to Other Abstractions¶
Current abstraction Poisson binomial distribution Domain-specific
Parents (1) — more general patterns this builds on
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Poisson binomial distribution is a kind of Probability Prime
Poisson Binomial Distribution instantiates Probability because it assigns normalized likelihoods to every possible success count under an explicitly stated random model.
Hierarchy paths (2) — routes to 2 parentless roots
- Poisson binomial distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Poisson binomial distribution → Probability → Measure → Set and Membership
Neighborhood in Abstraction Space¶
Poisson binomial distribution sits in a sparse region of the domain-specific corpus (90th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Delaporte Distribution — 0.80
- Negative Hypergeometric Distribution — 0.80
- Cumulant — 0.78
- Schuette–Nesbitt formula — 0.77
- Mixture Distribution — 0.77
Computed from structural-signature embeddings · 2026-09-08