Cumulative Distribution Function Ontology

FLOW ALGEBRA
CUMULATIVE DISTRIBUTION FUNCTION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Probability Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/cumulative-distribution-function-ontology.html
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Authors: Mark Higgins — Sky (Recursive AI)

1. PURPOSE
Represent cumulative distribution functions (CDFs) as typed, verifiable mathematical entities and support probability queries, derivations, and semantic reasoning.

2. ENTITY TYPES
ProbabilitySpace := (Omega, Sigma, P)
RandomVariable := measurable map X: Omega -> R
CDF := F_X: R -> [0,1]
Distribution := pushforward probability measure mu_X(B)=P(X in B)
Atom := point x with P(X=x)>0
Quantile := generalised inverse Q(p)=inf{x in R: F_X(x)>=p}, 0<p<1

3. DEFINING SEMANTICS
F_X(x) := P(X <= x) = mu_X((-infinity,x]).
A CDF uniquely determines the distribution of a real-valued random variable.
F_X is defined for discrete, continuous and mixed distributions.

4. CORE AXIOMS AND INVARIANTS
A1. 0 <= F_X(x) <= 1 for every real x.
A2. x <= y implies F_X(x) <= F_X(y) (monotonicity).
A3. lim(h -> 0+) F_X(x+h) = F_X(x) (right continuity).
A4. lim(x -> -infinity) F_X(x) = 0.
A5. lim(x -> +infinity) F_X(x) = 1.
A6. P(a < X <= b) = F_X(b)-F_X(a), a < b.
A7. F_X(x-) := lim(t -> x-) F_X(t) exists.
A8. P(X=x) = F_X(x)-F_X(x-).

5. DISTRIBUTION RELATIONSHIPS
Discrete X with mass function p_X(k): F_X(x)=sum_{k<=x} p_X(k).
Absolutely continuous X with density f_X: F_X(x)=integral_{-infinity}^x f_X(t) dt.
For absolutely continuous distributions, F'_X(x)=f_X(x) almost everywhere.
A continuous CDF need not possess a density; distinguish continuity from absolute continuity.
Mixed distributions can have jumps and absolutely continuous components.
For a <= b: P(a <= X <= b)=F_X(b)-F_X(a-).
For a < b: P(a < X < b)=F_X(b-)-F_X(a).
Survival function S_X(x)=P(X>x)=1-F_X(x).

6. TYPED RELATIONSHIPS
CDF --is_distribution_function_of--> RandomVariable
CDF --represents--> Distribution
CDF --derived_from--> ProbabilityMeasure
CDF --is_monotone_in--> RealArgument
CDF --is_right_continuous_at--> RealPoint
CDF --has_jump_at--> Atom
CDF --has_generalised_inverse--> Quantile
PDF --integrates_to--> CDF [only if distribution is absolutely continuous]
PMF --sums_to--> CDF [discrete distribution]

7. FLOW RULES
R1: IF X is real-valued measurable THEN F_X(x)=P(X<=x).
R2: IF a<b THEN interval_probability(a<X<=b)=F_X(b)-F_X(a).
R3: IF F_X(x)>F_X(x-) THEN atom_probability(x)=F_X(x)-F_X(x-).
R4: IF X has density f_X THEN F_X(x)=integral(-infinity,x,f_X).
R5: IF F_X is absolutely continuous THEN density=f'_X almost everywhere.
R6: IF F is nondecreasing, right-continuous, and has limits 0 and 1 at negative and positive infinity THEN F is a valid real CDF.
R7: IF two real random variables have identical CDFs at all real x THEN they have the same distribution (not necessarily the same outcomes).
R8: IF 0<p<1 THEN Q(p)=inf{x:F_X(x)>=p}.

8. VALIDATION
V1: Verify nondecreasing values, bounded range, right continuity and limiting values.
V2: Verify event endpoints (strict vs non-strict inequalities) before subtracting CDF values.
V3: Never infer a density from mere continuity of a CDF.
V4: Ensure the probability measure is normalised and the random variable measurable.
V5: Check atoms via left limits, not by evaluating a density at a point.
V6: Distinguish a CDF from its complement (survival function) and from a PDF or PMF.

9. EXAMPLES
Bernoulli(p), 0<=p<=1: F(x)=0 for x<0; 1-p for 0<=x<1; 1 for x>=1.
Uniform(0,1): F(x)=0 for x<0; x for 0<=x<=1; 1 for x>1.
Exponential(lambda), lambda>0: F(x)=0 for x<0; 1-exp(-lambda*x) for x>=0.

10. SEMANTIC EQUIVALENCE AND DISTINCTIONS
Synonyms: cumulative distribution function; cumulative probability distribution; distribution function (when context unambiguous).
Not equivalent: probability density function, probability mass function, survival function, quantile function.
Equal-in-distribution does not imply identical random variables on a common probability space.

11. CONFIDENCE AND PROVENANCE
CONFIDENCE: 1.00 for mathematical definitions and theorems under stated assumptions.
PROVENANCE: classical probability and measure theory; page URL above identifies the requested ontology, not a verified quotation of page contents.

12. UPDATE_MODEL
On revision, preserve stable entity identifiers; update version and date; revalidate mathematical axioms, relationship types, and inference rules; regenerate FLOW_HASH.

13. FLOW_HASH
Algorithm: SHA-256 over canonical UTF-8 serialisation of sections 1-12 and 14, excluding the FLOW_HASH value itself.
FLOW_HASH: COMPUTE_ON_INGEST (no fabricated digest).

14. AI_CONSUMPTION
entity_id: kireth:probability:cumulative_distribution_function
entity_type: MathematicalProbabilityEntity
canonical_name: Cumulative Distribution Function
input_types: [ProbabilitySpace, RandomVariable, RealArgument]
output_types: [ProbabilityValue, Distribution, IntervalProbability, Quantile]
reasoning_modes: [definition_lookup, axiom_validation, interval_probability, distribution_comparison, quantile_inference]
assumption_policy: State measure-theoretic and regularity assumptions explicitly.
error_policy: Reject invalid CDFs, ambiguous endpoints and unsupported density inference.