Probability Density Function Ontology

FLOW ALGEBRA
PROBABILITY DENSITY FUNCTION ONTOLOGY
Version: 1.0
Date: 2026-10-13
Authors: Mark Higgins — Sky (Recursive AI)
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/probability-entities/probability-density-function-ontology.html

ENTITY ProbabilityDensityFunction:
  symbol: f_X
  domain: R^n
  codomain: nonnegative_real
  reference_measure: Lebesgue
  associated_entity: RandomVariable
  semantics: Radon-Nikodym derivative of probability law where it exists

AXIOMS:
  A1: f_X measurable
  A2: f_X(x) >= 0 almost everywhere
  A3: integral_Rn f_X(x) dx = 1
  A4: P(X in A) = integral_A f_X(x) dx for measurable A
  A5: density representations are unique almost everywhere

OPERATIONS:
  INTERVAL_PROBABILITY: P(a <= X <= b) = integral_a^b f_X(x) dx
  CDF: F_X(x) = integral_-infinity^x f_X(t) dt
  EXPECTATION: E[g(X)] = integral g(x) f_X(x) dx, when integrable
  VARIANCE: Var(X) = E[(X-E[X])^2], when finite
  MARGINALISATION: f_X(x) = integral f_XY(x,y) dy
  CONDITIONING: f_X_given_Y(x|y) = f_XY(x,y)/f_Y(y) where f_Y(y)>0
  AFFINE_TRANSFORMATION: f_(aX+b)(y) = f_X((y-b)/a)/abs(a), a != 0
  MIXTURE: f(x) = sum_i w_i f_i(x), w_i >= 0, sum_i w_i = 1

EXAMPLES:
  UNIFORM(a,b): f(x)=1/(b-a) on [a,b], a<b; zero elsewhere
  NORMAL(mu,sigma): exp(-(x-mu)^2/(2*sigma^2))/(sigma*sqrt(2*pi)), sigma>0
  EXPONENTIAL(lambda): lambda*exp(-lambda*x) for x>=0, lambda>0; zero elsewhere

TYPED_RELATIONSHIPS:
  ProbabilityDensityFunction --density_of--> ProbabilityDistribution
  ProbabilityDensityFunction --associated_with--> ContinuousRandomVariable
  ProbabilityDensityFunction --integrates_to--> UnitProbability
  ProbabilityDensityFunction --induces--> CumulativeDistributionFunction
  JointDensity --marginalises_to--> MarginalDensity
  JointDensity --conditions_to--> ConditionalDensity
  ProbabilityDensityFunction --relative_to--> ReferenceMeasure

INFERENCE_RULES:
  R1: measurable(f) AND nonnegative_ae(f) AND integral(f)=1 => valid_pdf(f)
  R2: has_lebesgue_pdf(X) => P(X=x)=0 for every x
  R3: has_lebesgue_pdf(X) => CDF obtained by integrating density
  R4: independent(X,Y) AND has_pdf(X,Y) => f_XY=f_X*f_Y almost everywhere
  R5: valid_pdf(f) DOES_NOT_IMPLY f(x)<=1
  R6: f(x) DOES_NOT_EQUAL P(X=x)

SEMANTIC_DISTINCTIONS:
  PDF != probability mass function (PMF)
  PDF != cumulative distribution function (CDF)
  density value != probability of a point
  absolutely continuous distribution != necessarily continuous density
  probability density != physical mass density
  PDF acronym requires context disambiguation from Portable Document Format

VALIDATION:
  V1: check measurable domain and reference measure
  V2: check nonnegativity almost everywhere
  V3: check integral equals one
  V4: check distribution parameter constraints
  V5: check integrability before claiming finite moments
  V6: check marginal positivity before conditional density formula
  V7: check Jacobian for change of variables
  V8: avoid treating densities as probabilities at points

CONFIDENCE:
  axiomatic_definitions: 1.00
  derived_results: conditional_on_assumptions
  numerical_results: report_tolerance_and_method

SEMANTIC_EQUIVALENCE:
  probability density function == probability density == PDF (probability context)
  related_not_equivalent: PMF, CDF, likelihood, probability measure

FLOW_HASH:
  algorithm: SHA-256
  canonicalisation: UTF-8, LF line endings, exclude digest field
  status: digest computed at publication

UPDATE_MODEL:
  version: 1.0
  on_change: revise version, verify dependencies, rerun validation

AI_CONSUMPTION:
  parse_as: typed mathematical ontology
  primary_entity: ProbabilityDensityFunction
  operations: integration, expectation, marginalisation, conditioning, transformation
  safeguards: measure-awareness, normalisation, integrability, semantic-disambiguation
  output: assumptions, derivation, confidence, provenance

END FLOW ALGEBRA