Chapter 32
Random Variables as Distributed States

Flow Algebra

flow "Random Variables as Distributed States"

# ---------- CONCEPT ----------
# A random variable represents a distributed state — a measurable mapping that assigns values to uncertain outcomes.
# Through it, probability measure P on Ω becomes a distribution P_X on ℝ (or higher space).
# The random variable acts as a coherence map: it translates boundary-weighted uncertainty into value-weighted structure.

# ---------- STATES ----------
state RandomVariableSystem(R):
  SampleSpace:
    description: "Ω — underlying boundary field of all possible outcomes."
  SigmaAlgebra:
    description: "𝒜 — measurable subsets defining admissible boundaries."
  Measure:
    description: "P — probability weighting of each event A ∈ 𝒜."
  RandomVariable:
    description: "X: Ω → ℝⁿ — measurable mapping; assigns each outcome ω a numerical state X(ω)."
  Distribution:
    description: "P_X(B) = P(X ∈ B) — induced measure on ℝⁿ, the distribution of X."
  Density:
    description: "If absolutely continuous, f_X(x) = dP_X/dx — probability density function."
  Expectation:
    description: "E[X] = ∫ X dP = ∫ x f_X(x) dx — mean or coherent centroid of the variable."
  VarianceCovariance:
    description: "Var(X) = E[(X−E[X])²], Cov(X,Y)=E[(X−E[X])(Y−E[Y])]."
  Transformation:
    description: "Y = g(X) — new distributed state; distribution updated by change of variables."

# ---------- PARAMETERS ----------
params:
  measurability: "X^{-1}(B) ∈ 𝒜 for all Borel B"
  space: "ℝⁿ"
  integrability: "E[|X|] < ∞ when required"
  independence: "optional (X,Y independent ⇒ P_{X,Y}=P_XP_Y)"
  normalisation: "P_X(ℝⁿ)=1"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Distribution preservation: P_X(B) = P(X∈B)."
  - "Expectation linearity: E[aX+bY] = aE[X]+bE[Y]."
  - "Independence ⇒ joint factorisation: P_{X,Y}=P_XP_Y."
  - "Conditional expectation: E[X|𝔽] measurable w.r.t. σ-algebra 𝔽."
  - "Law of unconscious statistician (LOTUS): E[g(X)] = ∫ g(x) f_X(x) dx."
  - "Change of variables (1D): f_Y(y) = f_X(g^{-1}(y)) |(dg^{-1}/dy)|."
  - "Joint density integration: P(a<X≤b, c<Y≤d) = ∬_R f_{X,Y}(x,y) dx dy."
  - "Marginalisation: f_X(x) = ∫ f_{X,Y}(x,y) dy."
  - "Variance additive for independent: Var(X+Y)=Var(X)+Var(Y)."
  - "E[X²] = Var(X) + (E[X])²."

# ---------- TRANSFORMS ----------
transforms:
  Pushforward(X):
    rule: "P_X(B) = P(X∈B) — map probability from Ω to ℝ."
  ComputeExpectation(f_X):
    rule: "E[X] = ∫ x f_X(x) dx."
  ChangeOfVariable(g):
    rule: "If Y=g(X) with monotone g, then f_Y(y)=f_X(g^{-1}(y)) |(dg^{-1}/dy)|."
  ComputeVariance(f_X):
    rule: "Var(X)=∫ (x−μ)² f_X(x) dx."
  ConditionalDistribution(X|Y):
    rule: "f_{X|Y}(x|y)=f_{X,Y}(x,y)/f_Y(y) for f_Y(y)>0."
  IndependenceCheck(X,Y):
    rule: "Test f_{X,Y}(x,y)=f_X(x)f_Y(y)."
  JointToMarginal(f_{X,Y}):
    rule: "f_X(x)=∫ f_{X,Y}(x,y) dy; f_Y(y)=∫ f_{X,Y}(x,y) dx."
  ExpectationOfFunction(g):
    rule: "E[g(X)]=∫ g(x) f_X(x) dx (LOTUS)."
  MomentTransformations(X):
    rule: "Characteristic φ_X(t)=E[e^{itX}], MGF M_X(s)=E[e^{sX}] (when finite)."
  SumIndependent(X,Y):
    rule: "If indep., f_{X+Y}=f_X*f_Y (convolution)."

# ---------- MONITORS ----------
monitors:
  NormalisationCheck:
    measure: "∫ f_X(x) dx"
    target: 1
  MeanCheck:
    measure: "E[X] = ∫ x f_X(x) dx"
    meaning: "Consistency with empirical mean."
  VarianceCheck:
    measure: "E[(X−E[X])²]"
    meaning: "Ensures spread correctly computed."
  IndependenceTest:
    measure: "Δ = f_{X,Y}−f_X f_Y"
    meaning: "Δ≈0 ⇒ independence."
  CovarianceMonitor:
    measure: "Cov(X,Y)"
    meaning: "Positive ⇒ correlated; 0 ⇒ uncorrelated."
  TransformCheck:
    meaning: "For Y=g(X), check ∫ f_Y(y) dy=1 (density preserved)."
  ConditionalConsistency:
    meaning: "Ensure f_{X|Y}(x|y)f_Y(y)=f_{X,Y}(x,y)."

# ---------- EQUATIONS ----------
equations:
  - "P_X(B)=P(X∈B)"
  - "E[X]=\\int_{\\mathbb{R}} x f_X(x) dx"
  - "Var(X)=E[X^2]−(E[X])^2"
  - "Cov(X,Y)=E[XY]−E[X]E[Y]"
  - "E[g(X)]=\\int g(x) f_X(x) dx"
  - "f_Y(y)=f_X(g^{-1}(y)) |\\frac{dg^{-1}}{dy}|"
  - "f_{X+Y}(z)=\\int f_X(x)f_Y(z−x) dx"
  - "φ_X(t)=E[e^{itX}],\\; φ_{X+Y}(t)=φ_X(t)φ_Y(t) (if indep.)"
  - "f_X(x)=\\int f_{X,Y}(x,y) dy"
  - "f_{X|Y}(x|y)=\\frac{f_{X,Y}(x,y)}{f_Y(y)}"

# ---------- NOTES ----------
notes:
  - "A random variable is a measurable projection of uncertainty into value space — it assigns numbers to outcomes coherently."
  - "Its distribution is the shadow of probability cast onto ℝ: the pushforward of measure through X."
  - "Expectation, variance, and covariance describe centrality, spread, and joint coherence of the distribution."
  - "Transformations propagate uncertainty — when X changes via g, density rebalances through Jacobian scaling."
  - "Independence expresses separability of coherence: joint density factorises into individual flows."
  - "Moments and transforms (characteristic, MGF) encode the entire distribution into algebraic form."
  - "In Flow Algebra, random variables represent distributed states — regions of uncertainty projected into measurable form. Their flow evolves through expectation, transformation, and dependence structure."