Random Variable Ontology

FULL EXECUTABLE FLOW ALGEBRA — RANDOM VARIABLE ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/foundations-of-probability/random-variable-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Foundations of Probability"
ONTOLOGY := "Random Variable Ontology"
PRIMARY_ENTITY := RandomVariable
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

RandomVariable :=
    MeasurableFunction
    ∧ MapsOutcomesToValues
    ∧ DefinedOnProbabilitySpace
    ∧ InducesProbabilityDistribution

ProbabilitySpace :=
    (Ω,F,P)

RandomVariableDefinition :=
    X : Ω → S

RealValuedRandomVariable :=
    X : Ω → R

Measurability :=
    ∀B ∈ B(S) :
        X^{-1}(B) ∈ F

Outcome :=
    ω ∈ Ω

RandomVariableValue :=
    X(ω)

StateSpace :=
    S

Support :=
    ValuesWithPositiveProbabilityOrDensitySupport
    under relevant distribution definition

DiscreteRandomVariable :=
    RandomVariable
    ∧ CountableRange

ContinuousRandomVariable :=
    RandomVariable
    ∧ DistributionTypicallyRepresentedByDensity
    where density exists

MixedRandomVariable :=
    Distribution
    containing
    DiscreteAndContinuousComponents

DegenerateRandomVariable :=
    ∃c :
        P(X=c)=1

BernoulliRandomVariable :=
    X∈{0,1}
    ∧ P(X=1)=p

IndicatorRandomVariable :=
    1_A(ω)
    =
    {
        1 if ω∈A,
        0 otherwise
    }

IndicatorExpectation :=
    E[1_A]=P(A)

Distribution :=
    Law(X)
    :=
    P_X

PushforwardMeasure :=
    P_X(B)
    =
    P(X^{-1}(B))

CDF :=
    F_X(x)
    =
    P(X≤x)

CDFProperties := {
    NonDecreasing,
    RightContinuous,
    LimitMinusInfinity=0,
    LimitPlusInfinity=1
}

PMF :=
    p_X(x)
    =
    P(X=x)

PMFConstraints :=
    p_X(x)≥0
    ∧ Σ_x p_X(x)=1

PDF :=
    f_X(x)

PDFConstraints :=
    f_X(x)≥0
    ∧ Integral f_X(x)dx=1

DensityProbability :=
    P(a≤X≤b)
    =
    Integral_a^b f_X(x)dx

PointProbabilityContinuous :=
    AbsolutelyContinuous(X)
    ⇒ P(X=x)=0

QuantileFunction :=
    Q(p)
    =
    inf{x : F_X(x)≥p}

Median :=
    m
    satisfying
    P(X≤m)≥1/2
    ∧ P(X≥m)≥1/2

Mode :=
    ValueOrRegion
    maximising
    PMFOrDensity
    under relevant definition

Expectation :=
    E[X]
    =
    Integral_Ω X dP
    when integrable

DiscreteExpectation :=
    E[X]
    =
    Σ_x x p_X(x)

ContinuousExpectation :=
    E[X]
    =
    Integral x f_X(x)dx

Integrability :=
    E[|X|] < ∞

LinearityOfExpectation :=
    E[aX+bY]
    =
    aE[X]+bE[Y]

ExpectedFunction :=
    E[g(X)]
    =
    Integral g(X)dP

LOTUSDiscrete :=
    E[g(X)]
    =
    Σ_x g(x)p_X(x)

LOTUSContinuous :=
    E[g(X)]
    =
    Integral g(x)f_X(x)dx

Moment :=
    E[X^k]

RawMoment :=
    μ'_k
    =
    E[X^k]

CentralMoment :=
    μ_k
    =
    E[(X-E[X])^k]

Variance :=
    Var(X)
    =
    E[(X-E[X])²]

VarianceIdentity :=
    Var(X)
    =
    E[X²]-(E[X])²

StandardDeviation :=
    SD(X)
    =
    sqrt(Var(X))

VarianceScaling :=
    Var(aX+b)
    =
    a²Var(X)

Skewness :=
    StandardisedThirdCentralMoment

Kurtosis :=
    StandardisedFourthCentralMoment

MomentGeneratingFunction :=
    M_X(t)
    =
    E[e^(tX)]
    where finite near relevant t

CharacteristicFunction :=
    φ_X(t)
    =
    E[e^(itX)]

ProbabilityGeneratingFunction :=
    G_X(s)
    =
    E[s^X]
    for suitable nonnegative integer-valued X

TransformUniqueness :=
    CharacteristicFunction
    uniquely_determines
    Distribution

RandomVector :=
    X = (X1,...,Xn)

JointDistribution :=
    Law(X1,...,Xn)

JointCDF :=
    F(x1,...,xn)
    =
    P(X1≤x1,...,Xn≤xn)

JointPMF :=
    p(x,y)
    =
    P(X=x,Y=y)

JointPDF :=
    f_(X,Y)(x,y)

MarginalDistribution :=
    JointDistribution
    → SumOrIntegrateOtherVariables

ConditionalDistribution :=
    Law(X|Y)

ConditionalPMF :=
    P(X=x|Y=y)

ConditionalDensity :=
    f_(X|Y)(x|y)
    =
    f_(X,Y)(x,y)/f_Y(y)
    where denominator positive

Independence :=
    X ⟂ Y
    iff
    P(X∈A,Y∈B)
    =
    P(X∈A)P(Y∈B)
    for all measurable A,B

DiscreteIndependence :=
    p_(X,Y)(x,y)
    =
    p_X(x)p_Y(y)

ContinuousIndependence :=
    f_(X,Y)(x,y)
    =
    f_X(x)f_Y(y)
    almost everywhere
    where joint density exists

ConditionalIndependence :=
    X ⟂ Y | Z

Covariance :=
    Cov(X,Y)
    =
    E[(X-E[X])(Y-E[Y])]

CovarianceIdentity :=
    Cov(X,Y)
    =
    E[XY]-E[X]E[Y]

Correlation :=
    Corr(X,Y)
    =
    Cov(X,Y)/(SD(X)SD(Y))
    when standard deviations positive

IndependentImpliesZeroCovariance :=
    Independence(X,Y)
    ∧ finite_second_moments
    ⇒ Cov(X,Y)=0

ZeroCovarianceDoesNotGenerallyImplyIndependence :=
    Cov(X,Y)=0
    ↛ X ⟂ Y

Transformation :=
    Y=g(X)

TransformedRandomVariable :=
    g∘X

DiscreteTransformation :=
    p_Y(y)
    =
    Σ_{x:g(x)=y} p_X(x)

MonotoneContinuousTransformation :=
    Y=g(X)
    ∧ invertible_monotone(g)
    ⇒
    f_Y(y)
    =
    f_X(g^{-1}(y))
    |d/dy g^{-1}(y)|

MultivariateTransformation :=
    Y=g(X)
    → JacobianTransformation
    under invertibility conditions

Jacobian :=
    determinant(
        derivative_matrix
    )

SumOfRandomVariables :=
    S=X+Y

ConvolutionDiscrete :=
    Independent(X,Y)
    ⇒
    p_S(s)
    =
    Σ_x p_X(x)p_Y(s-x)

ConvolutionContinuous :=
    Independent(X,Y)
    ⇒
    f_S(s)
    =
    Integral f_X(x)f_Y(s-x)dx

LinearCombination :=
    Y
    =
    a1X1+...+anXn

ExpectationLinearCombination :=
    E[Y]
    =
    Σ_i a_iE[X_i]

VarianceLinearCombination :=
    Var(Σ_i a_iX_i)
    =
    Σ_i a_i²Var(X_i)
    +
    2Σ_{i<j}a_i a_j Cov(X_i,X_j)

IndependentVarianceSum :=
    PairwiseIndependent(X_i)
    ⇒
    Var(Σ_i X_i)
    =
    Σ_i Var(X_i)

ConditionalExpectation :=
    E[X|Y]
    ∨ E[X|G]

ConditionalExpectationAsRandomVariable :=
    E[X|G]
    is G-measurable

TowerProperty :=
    H⊆G
    ⇒
    E[E[X|G]|H]
    =
    E[X|H]

LawOfTotalExpectation :=
    E[X]
    =
    E[E[X|Y]]

LawOfTotalVariance :=
    Var(X)
    =
    E[Var(X|Y)]
    +
    Var(E[X|Y])

RandomVariableSequence :=
    {X_n}

IIDSequence :=
    Independent(X_n)
    ∧ IdenticallyDistributed(X_n)

IdenticallyDistributed :=
    Law(X_i)=Law(X_j)
    for all i,j

AlmostSureConvergence :=
    X_n → X a.s.
    iff
    P({ω:X_n(ω)→X(ω)})=1

ConvergenceInProbability :=
    ∀ε>0 :
        P(|X_n-X|>ε)→0

ConvergenceInDistribution :=
    F_(X_n)(x)
    → F_X(x)
    at continuity points of F_X

MeanSquareConvergence :=
    E[(X_n-X)²]→0

ConvergenceRelations := {
    MeanSquareConvergence ⇒ ConvergenceInProbability,
    AlmostSureConvergence ⇒ ConvergenceInProbability,
    ConvergenceInProbability ⇒ ConvergenceInDistribution
}

SampleMean :=
    X_bar_n
    =
    (1/n)Σ_{i=1..n}X_i

WeakLawOfLargeNumbers :=
    X_bar_n
    → E[X]
    in probability
    under theorem conditions

StrongLawOfLargeNumbers :=
    X_bar_n
    → E[X]
    almost surely
    under theorem conditions

CentralLimitTheorem :=
    ProperlyStandardisedSum
    → StandardNormal
    in distribution
    under theorem conditions

OrderStatistic :=
    X_(1)≤...≤X_(n)

Minimum :=
    X_(1)

Maximum :=
    X_(n)

SampleMedian :=
    MiddleOrderStatistic
    under sample-size convention

RandomVariableConstraint :=
    ConstraintOn(
        Support,
        Distribution,
        Moments,
        Dependence,
        Transformation
    )

DistributionConstraint :=
    Law(X)
    must_satisfy
    ProbabilityAxioms

MomentConstraint :=
    SpecifiedExpectationOrVariance

SupportConstraint :=
    X∈S
    almost surely

RandomVariableConsistency :=
    Distribution
    ∧ Support
    ∧ Moments
    ∧ DependenceRelations
    jointly_compatible

InvalidRandomVariableModel :=
    Violates(
        Measurability
        ∨ ProbabilityAxioms
        ∨ DistributionNormalisation
        ∨ ActiveConstraints
    )

RandomVariableEvidence :=
    Observation
    ∨ SampleData
    ∨ ModelAssumption
    ∨ DistributionSpecification

RandomVariableState := {
    ProbabilitySpace,
    VariableDefinition,
    Support,
    Distribution,
    Moments,
    DependenceRelations,
    Constraints,
    Evidence,
    Confidence,
    Provenance
}

RandomVariableUpdate :=
    NewEvidence
    → UpdateDistributionOrParameters
    → RecomputeMoments
    → ReevaluateDependence
    → RecheckConstraints

SEMANTIC_EQUIVALENCE(X,Y) :=
    same(Distribution)
    under specified semantic scope
    ∧ same(RelevantDependenceRelations)
    where required

EqualInDistribution :=
    X =_d Y
    iff
    Law(X)=Law(Y)

AlmostSureEquality :=
    X=Y a.s.
    iff
    P(X=Y)=1

EqualityInProbabilityModel :=
    Distinguish(
        PointwiseEquality,
        AlmostSureEquality,
        EqualityInDistribution
    )

RandomVariableOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ InferenceRules

CoreClasses := {
    RandomVariable,
    ProbabilitySpace,
    SampleSpace,
    Outcome,
    MeasurableFunction,
    Support,
    Distribution,
    PMF,
    PDF,
    CDF,
    Expectation,
    Variance,
    Moment,
    RandomVector,
    JointDistribution,
    ConditionalDistribution,
    Independence,
    Covariance,
    Transformation,
    RandomVariableSequence
}

CoreRelations := {
    mapsFrom,
    mapsTo,
    measurableWithRespectTo,
    distributedAs,
    hasSupport,
    hasExpectation,
    hasVariance,
    hasMoment,
    conditionedOn,
    independentOf,
    covariesWith,
    transformedBy,
    convergesTo,
    derivedFrom,
    dependsOn,
    tracesTo
}

TypedRelations :=
    RandomVariable --mapsFrom--> SampleSpace
    RandomVariable --mapsTo--> StateSpace
    RandomVariable --measurableWithRespectTo--> SigmaAlgebra
    RandomVariable --distributedAs--> Distribution
    RandomVariable --hasSupport--> Support
    RandomVariable --hasExpectation--> ExpectedValue
    RandomVariable --hasVariance--> Variance
    ConditionalDistribution --conditionedOn--> RandomVariable
    RandomVariable --independentOf--> RandomVariable
    RandomVariable --transformedBy--> Function
    RandomVariableSequence --convergesTo--> RandomVariable

InferenceRules := {
    Measurable(X)
        ⇒ DistributionExists(X),

    PMF(p_X)
        ⇒ Σ_x p_X(x)=1,

    PDF(f_X)
        ⇒ Integral f_X(x)dx=1,

    Integrable(X)
        ⇒ ExpectationExistsFinite(X),

    Independent(X,Y)
        ∧ finite_second_moments
        ⇒ Cov(X,Y)=0,

    Y=g(X)
        ⇒ Y is RandomVariable
        when g is suitably measurable,

    MeanSquareConvergence(X_n,X)
        ⇒ ConvergenceInProbability(X_n,X),

    ConvergenceInProbability(X_n,X)
        ⇒ ConvergenceInDistribution(X_n,X)
}

CanonicalRandomVariableRepresentation :=
    ParseVariableDefinition
    → ResolveProbabilitySpace
    → ResolveMeasurability
    → ResolveSupport
    → ResolveDistribution
    → ResolveMoments
    → ResolveDependence
    → ResolveTransformations
    → ValidateConstraints

RandomVariableGraph :=
    G=(V,E)

V := {
    ProbabilitySpaces,
    Outcomes,
    RandomVariables,
    Supports,
    Distributions,
    PMFs,
    PDFs,
    CDFs,
    Moments,
    Expectations,
    Variances,
    DependenceRelations,
    Transformations,
    ConvergenceRelations
}

E := {
    mapsFrom,
    mapsTo,
    measurableWithRespectTo,
    distributedAs,
    hasSupport,
    hasExpectation,
    hasVariance,
    conditionedOn,
    independentOf,
    covariesWith,
    transformedBy,
    convergesTo,
    dependsOn
}

CONFIDENCE(RandomVariableInference) :=
    weighted_mean(
        VariableDefinitionQuality,
        ProbabilitySpaceResolution,
        DistributionResolution,
        ParameterEvidence,
        DependenceResolution,
        ConstraintConsistency,
        CalculationValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectRandomVariableConcept
    → IdentifyVariable
    → ResolveProbabilitySpace
    → ResolveDistributionInformation
    → ResolveRequestedProperty
    → IdentifyRelevantConstraints

AI_REASONING_FLOW :=
    Query
    → ParseRandomVariableProblem
    → ResolveProbabilitySpace
    → ResolveVariableMapping
    → CheckMeasurabilityWhereRelevant
    → ResolveSupport
    → ResolveDistribution
    → DetermineDiscreteContinuousOrMixedType
    → ComputeRequestedProbabilityOrMoment
    → ResolveJointOrConditionalStructure
    → EvaluateDependence
    → ApplyTransformationsIfRequired
    → EvaluateConvergenceIfRequired
    → ValidateConstraints
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseRandomVariableEntities,
    ParseTypedRelations,
    ResolveProbabilitySpaces,
    ResolveMeasurability,
    ResolveSupports,
    ResolveDistributions,
    EvaluatePMFs,
    EvaluatePDFs,
    EvaluateCDFs,
    ComputeExpectations,
    ComputeVariances,
    ComputeMoments,
    ResolveJointDistributions,
    ResolveConditionalDistributions,
    EvaluateIndependence,
    ApplyTransformations,
    EvaluateConvergence,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewObservationDistributionOrConstraint
    → ValidateInput
    → IdentifyAffectedRandomVariables
    → UpdateDistributionState
    → RecomputeProbabilities
    → RecomputeMoments
    → ReevaluateDependence
    → ReevaluateTransformations
    → RecheckConstraints
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DistributionUpdate :=
    ChangedDistribution
    ⇒ recompute(
        CDF,
        Expectation,
        Variance,
        Moments,
        TailProbabilities
    )

DependenceUpdate :=
    ChangedJointOrConditionalInformation
    ⇒ recompute(
        Covariance,
        Correlation,
        IndependenceStatus
    )

TransformationUpdate :=
    ChangedTransformation
    ⇒ recompute(
        TransformedDistribution,
        TransformedMoments
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        RandomVariableDefinition,
        Distribution,
        Expectation,
        Independence,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    RandomVariableEntityResolution,
    ProbabilitySpace,
    VariableMapping,
    MeasurabilityStatus,
    VariableType,
    Support,
    Distribution,
    ProbabilityFunctions,
    Moments,
    DependenceStructure,
    TransformationStructure,
    ConvergenceStatus,
    ModelConsistency,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA