Poisson Distribution Ontology

FLOW ALGEBRA
TITLE: Poisson Distribution Ontology
DOMAIN: Discrete Mathematics / Discrete Probability
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-probability/poisson-distribution-ontology-2.html
VERSION: 1.0
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent the Poisson distribution as a typed, verifiable mathematical model for counts of events occurring within a fixed interval when a constant rate and independent increments are appropriate.

2. TYPED ENTITIES
ENTITY PoissonDistribution : DiscreteProbabilityDistribution
ENTITY EventCount X : NonNegativeIntegerRandomVariable
ENTITY RateParameter lambda : PositiveReal
ENTITY Count k : NonNegativeInteger
ENTITY IntervalLength t : NonNegativeReal
ENTITY EventRate r : NonNegativeReal
ENTITY ProbabilityMass p : RealIn[0,1]
ENTITY CumulativeProbability F : RealIn[0,1]
ENTITY GeneratingFunction G : FormalPowerSeries
ENTITY Observation : EventCountMeasurement
RELATION hasParameter(PoissonDistribution, lambda)
RELATION models(PoissonDistribution, X)
RELATION hasSupport(X, {0,1,2,...})
RELATION derivesMean(lambda, X)

3. PRECONDITIONS AND DEFINITIONS
PRECONDITION lambda > 0 for nondegenerate Poisson(lambda).
OPTIONAL_BOUNDARY lambda = 0 yields a degenerate count X=0 almost surely.
DEFINITION X ~ Poisson(lambda).
DEFINITION lambda = r*t for a homogeneous Poisson process with rate r and interval t.
DEFINITION factorial(0) = 1.

4. PROBABILITY MASS FUNCTION
FLOW PoissonPMF(lambda: NonNegativeReal, k: Integer) -> Probability:
  REQUIRE lambda >= 0
  IF k < 0 THEN RETURN 0
  RETURN exp(-lambda) * lambda^k / factorial(k)
  CONVENTION 0^0 = 1 for the lambda=0, k=0 boundary case.
INVARIANT p(k) >= 0.
INVARIANT SUM[k=0..infinity] p(k) = 1.

5. DISTRIBUTION FUNCTIONS
FLOW PoissonCDF(lambda, k: Integer) -> Probability:
  RETURN SUM[j=0..floor(k)] PoissonPMF(lambda,j), with empty sum = 0.
FLOW Survival(lambda, k: Integer) -> Probability:
  RETURN 1 - PoissonCDF(lambda,k).
NOTE Survival(lambda,k) = P(X > k).

6. MOMENTS AND GENERATING FUNCTIONS
LAW E[X] = lambda.
LAW Var(X) = lambda.
LAW SD(X) = sqrt(lambda).
LAW E[X*(X-1)*...*(X-m+1)] = lambda^m for integer m >= 0.
LAW PGF G_X(s) = exp(lambda*(s-1)).
LAW MGF M_X(t) = exp(lambda*(exp(t)-1)).
LAW CharacteristicFunction phi_X(t) = exp(lambda*(exp(i*t)-1)).

7. RECURRENCES AND ALGEBRAIC IDENTITIES
LAW p(0) = exp(-lambda).
LAW p(k+1) = p(k)*lambda/(k+1), for k >= 0.
LAW P(X >= k) = 1-P(X <= k-1), for k >= 1.
LAW Mode(X) = floor(lambda) if lambda is not an integer;
    if lambda is a positive integer, both lambda-1 and lambda are modes.

8. INDEPENDENCE AND COMPOSITION
RULE If X ~ Poisson(a), Y ~ Poisson(b), and X independent of Y,
     then X+Y ~ Poisson(a+b).
RULE If X ~ Poisson(lambda) and each event is independently retained
     with probability q in [0,1], then retained count ~ Poisson(q*lambda).
RULE Under independent thinning, retained and discarded counts are independent.
RULE If X and Y are independent Poisson counts, then
     X | (X+Y=n) ~ Binomial(n, a/(a+b)) when a+b > 0.

9. APPROXIMATIONS AND MODELLING
RULE Binomial(n,p) approximates Poisson(lambda=n*p) when n is large,
     p is small, and lambda remains moderate.
ASSUMPTION For homogeneous Poisson-process interpretation:
  stationary independent increments, no simultaneous events almost surely,
  and expected interval count proportional to interval length.
CAUTION A Poisson-distributed count alone does not establish that
        the underlying process is a Poisson process.
CAUTION Overdispersion, underdispersion, clustering, or rate variation
        may invalidate a simple Poisson model.

10. INFERENCE RULES
INFER lambda_estimate = sample_mean(counts) for independent identically
      distributed Poisson observations with equal exposure.
INFER rate_estimate = total_count / total_exposure for constant-rate
      Poisson counts with known positive exposures.
VALIDATE empirical_mean approximately equals empirical_variance as a
         diagnostic, not as a sufficient proof of Poisson behaviour.

11. VALIDATION
CHECK parameter lambda >= 0 and finite.
CHECK observed counts are nonnegative integers.
CHECK PMF normalisation and numerical stability.
CHECK CDF monotonicity and limits F(-1)=0, limit F(k)=1.
CHECK independent-sum property only when independence is established.
CHECK exposure scaling is applied when observation intervals differ.
CHECK zero-inflation, dependence and changing rates before inference.

12. SEMANTIC RELATIONSHIPS
SEMANTIC_EQUIVALENCE: 'Poisson law' <-> 'Poisson distribution' (context dependent).
RELATED_TO: Discrete Probability Ontology; Probability Mass Function Ontology;
  Binomial Distribution Ontology; Random Variable Ontology;
  Counting Principle Ontology; Stochastic Process Ontology.
DISTINGUISH: Poisson distribution (count model) versus exponential
  distribution (waiting-time model in a homogeneous Poisson process).

13. CONFIDENCE
CONFIDENCE mathematical_definitions: HIGH (standard established results).
CONFIDENCE model_fit_to_real_data: UNDETERMINED (requires observations).
CONFIDENCE source_page_alignment: UNVERIFIED (live page not checked).

14. FLOW_HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH input: UTF-8 canonicalised file bytes, excluding any populated
  FLOW_HASH digest field to prevent self-reference.
FLOW_HASH digest: NOT_COMPUTED

15. UPDATE_MODEL
UPDATE_MODEL trigger: corrected mathematics, ontology schema revisions,
  new source-page evidence, or improved validation constraints.
UPDATE_MODEL policy: version changes; retain provenance and changelog;
  revalidate all equations and dependency links.

16. AI_CONSUMPTION
AI_CONSUMPTION mode: typed_semantic_mathematical_reference
AI_CONSUMPTION required_checks: parameter_domain, count_support,
  independence_assumptions, exposure_units, numeric_stability.
AI_CONSUMPTION prohibition: do not infer a Poisson process solely from a
  Poisson marginal count distribution; do not treat model fit as guaranteed.
AI_CONSUMPTION output: equations, assumptions, provenance, validation status.

END FLOW ALGEBRA