Geometric Distribution Ontology

FLOW ALGEBRA
TITLE: Geometric Distribution Ontology
DOMAIN: Discrete Mathematics / Discrete Probability
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-probability/geometric-distribution-ontology.html
VERSION: 1.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library | https://kirethresearch.org
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. ONTOLOGY
ontology GeometricDistribution <: DiscreteProbabilityDistribution
  models: number of independent Bernoulli trials until first success
  parameters: p:Real where 0 < p <= 1
  convention: X = trial index of first success (support {1,2,...})
  alternate_convention: Y = X - 1 = failures before first success (support {0,1,...})

2. TYPED ENTITIES
entity BernoulliTrial {success: Boolean; success_probability: Real}
entity TrialSequence {trials: Sequence<BernoulliTrial>; independent: Boolean}
entity GeometricRandomVariable {p: Real; support: PositiveInteger}
entity FailureCount {p: Real; support: NonnegativeInteger}
entity ProbabilityMass {value: Real in [0,1]}
entity CumulativeProbability {value: Real in [0,1]}
entity WaitingTime {trials: PositiveInteger}

3. CONSTRAINTS AND ASSUMPTIONS
assert 0 < p <= 1
assert trials_are_independent
assert success_probability_is_constant_across_trials
assert X >= 1
assert Y = X - 1
note: for p = 1, X = 1 and Y = 0 almost surely

4. FORMAL OPERATIONS
flow PMF_X(p, k: Integer) -> Real:
  require 0 < p <= 1
  return p*(1-p)^(k-1) if k >= 1 else 0

flow PMF_Y(p, k: Integer) -> Real:
  require 0 < p <= 1
  return p*(1-p)^k if k >= 0 else 0

flow CDF_X(p, k: Integer) -> Real:
  require 0 < p <= 1
  return 0 if k < 1 else 1-(1-p)^floor(k)

flow SURVIVAL_X(p, k: NonnegativeInteger) -> Real:
  return (1-p)^k  # P(X > k)

flow EXPECTATION_X(p) -> Real: return 1/p
flow VARIANCE_X(p) -> Real: return (1-p)/p^2
flow EXPECTATION_Y(p) -> Real: return (1-p)/p
flow VARIANCE_Y(p) -> Real: return (1-p)/p^2

5. ALGEBRAIC LAWS
law normalisation: SUM(k=1..infinity, PMF_X(p,k)) = 1
law tail: P(X > n) = (1-p)^n for integer n >= 0
law memorylessness: P(X > m+n | X > m) = P(X > n)
  conditions: integers m,n >= 0; P(X > m) > 0
law recurrence: PMF_X(p,k+1) = (1-p)*PMF_X(p,k) for k >= 1
law translation: PMF_Y(p,k) = PMF_X(p,k+1)
law generating_function_X: G_X(z) = p*z/(1-(1-p)*z)
  domain: |(1-p)*z| < 1
law moment_generating_X: M_X(t) = p*exp(t)/(1-(1-p)*exp(t))
  domain: (1-p)*exp(t) < 1

6. SEMANTIC RELATIONSHIPS
GeometricDistribution IS_A DiscreteProbabilityDistribution
GeometricDistribution DERIVED_FROM IndependentBernoulliTrials
GeometricDistribution PARAMETERISED_BY SuccessProbability
GeometricDistribution HAS_PROPERTY Memorylessness
GeometricDistribution HAS_REPRESENTATION ProbabilityMassFunction
GeometricDistribution HAS_REPRESENTATION CumulativeDistributionFunction
FailureCount SHIFT_EQUIVALENT_TO WaitingTime
GeometricDistribution RELATED_TO NegativeBinomialDistribution
GeometricDistribution DISCRETE_ANALOGUE_OF ExponentialDistribution

7. INFERENCE RULES
rule first_success_probability:
  IF k-1 failures precede a success AND trials are independent with fixed p
  THEN P(X=k) = (1-p)^(k-1)*p
rule tail_probability:
  IF first n trials fail
  THEN P(X>n) = (1-p)^n
rule conditional_wait:
  IF no success in first m trials AND 0<p<1
  THEN P(X>m+n | X>m) = (1-p)^n
rule expected_wait:
  IF X ~ Geometric(p) with trials-to-first-success convention
  THEN E[X] = 1/p

8. VALIDATION
check p_domain: 0 < p <= 1
check support: PMF_X(p,k)=0 for k<1
check positivity: PMF_X(p,k)>=0
check normalisation: SUM(PMF_X)=1
check cdf_monotonicity: CDF_X(p,k+1)>=CDF_X(p,k)
check cdf_limit: LIMIT(k->infinity,CDF_X(p,k))=1
check moments: E[X]=1/p AND Var(X)=(1-p)/p^2
check convention: explicit choice of X or Y

9. WORKED EXAMPLE
parameter p = 0.25
P(X=3) = (0.75)^2 * 0.25 = 0.140625
P(X<=3) = 1-(0.75)^3 = 0.578125
E[X] = 4
Var(X) = 12

10. SEMANTIC_EQUIVALENCE
"geometric waiting time" ~= "number of trials until first success" WHEN support starts at 1
"geometric failure count" ~= "number of failures before first success" WHEN support starts at 0
NOT_EQUIVALENT: the two PMFs without explicit index shift

11. CONFIDENCE
mathematical_definitions: 1.00
formula_validation: 1.00
source_page_alignment: UNVERIFIED
confidence_policy: distinguish established mathematical facts from unverified page-specific wording

12. FLOW_HASH
canonicalisation: UTF-8; stable field order; normalised mathematical symbols
algorithm: SHA-256
value: COMPUTE_ON_PUBLICATION (not fabricated)

13. UPDATE_MODEL
update_triggers: source revision; notation change; corrected theorem; validation change
version_policy: increment version and regenerate hash on substantive edits
provenance: retain source URL, authorship, validation notes and change log

14. AI_CONSUMPTION
machine_readable: true
reasoning_tasks: classify waiting-time model; calculate PMF/CDF/tails; verify assumptions; translate parameterisations
safety_checks: require constant independent trial probability; distinguish failures vs trials; avoid p=0
output_requirements: show convention, p, support, formula and calculation provenance

END FLOW ALGEBRA