Chapter 33
Expectation and Variance as Field Averages

Flow Algebra

flow "Expectation and Variance as Field Averages"

# ---------- CONCEPT ----------
# Expectation represents the centre of coherence in a probability field — the weighted balance of outcomes.
# Variance measures the spread of coherence — how far the field extends from its mean.
# Together, they define the first and second moments of the probability distribution: balance and dispersion in the space of uncertainty.

# ---------- STATES ----------
state MomentSystem(M):
  RandomVariable:
    description: "X: Ω → ℝ — measurable distributed state."
  Distribution:
    description: "P_X or density f_X(x) — weight field describing probability distribution."
  Expectation:
    description: "E[X] = ∫ x f_X(x) dx — mean or centre of coherence."
  Variance:
    description: "Var(X) = E[(X−E[X])²] — field spread about the mean."
  HigherMoments:
    description: "μ_n = E[(X−E[X])^n] — generalised shape descriptors of distribution."
  Covariance:
    description: "Cov(X,Y) = E[(X−E[X])(Y−E[Y])] — joint coherence measure."
  Correlation:
    description: "ρ(X,Y) = Cov(X,Y)/(σ_X σ_Y) — normalised alignment of fluctuations."

# ---------- PARAMETERS ----------
params:
  density: "f_X(x) ≥ 0, ∫ f_X(x) dx = 1"
  measure_space: "(Ω, 𝒜, P)"
  integrability: "E[|X|] < ∞, E[X²] < ∞"
  normalization: "P(Ω)=1"
  independence: optional
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "E[X] = ∫ x f_X(x) dx = ∫_Ω X dP."
  - "E[aX + b] = aE[X] + b."
  - "Var(X) = E[X²] − (E[X])²."
  - "Var(aX + b) = a² Var(X)."
  - "Cov(X,Y) = E[XY] − E[X]E[Y]."
  - "Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)."
  - "If X, Y independent ⇒ Cov(X,Y)=0 ⇒ Var(X+Y)=Var(X)+Var(Y)."
  - "E[g(X)] = ∫ g(x) f_X(x) dx (LOTUS)."
  - "Law of total expectation: E[X] = E[E[X|Y]]."
  - "Law of total variance: Var(X) = E[Var(X|Y)] + Var(E[X|Y])."

# ---------- TRANSFORMS ----------
transforms:
  ComputeExpectation(f_X):
    rule: "E[X] = ∫ x f_X(x) dx."
  ComputeVariance(f_X):
    rule: "Var(X) = ∫ (x−μ)² f_X(x) dx."
  ShiftVariable(Y=aX+b):
    rule: "E[Y]=aE[X]+b, Var(Y)=a²Var(X)."
  JointVariance(X,Y):
    rule: "Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)."
  CovarianceCalc(f_{X,Y}):
    rule: "Cov(X,Y)=∬ (x−μ_X)(y−μ_Y) f_{X,Y}(x,y) dx dy."
  ExpectFunction(g):
    rule: "E[g(X)]=∫ g(x) f_X(x) dx."
  ConditionalExpectation(X|Y):
    rule: "E[X|Y]=∫ x f_{X|Y}(x|Y) dx — expectation as function of Y."
  TotalVarianceDecompose(X,Y):
    rule: "Var(X)=E[Var(X|Y)]+Var(E[X|Y])."
  CenterVariable(X):
    rule: "X_c = X − E[X] — mean-removed variable."

# ---------- MONITORS ----------
monitors:
  NormalisationCheck:
    measure: "∫ f_X(x) dx"
    target: 1
  MeanZeroCheck:
    measure: "E[X_c]"
    meaning: "Should equal 0 after centering."
  VarianceNonnegative:
    measure: "Var(X)"
    condition: "Var(X) ≥ 0 with equality iff X constant."
  CovarianceSymmetry:
    measure: "Cov(X,Y)−Cov(Y,X)"
    meaning: "Zero ⇒ symmetry holds."
  CorrelationBounds:
    measure: "ρ(X,Y)"
    condition: "−1 ≤ ρ ≤ 1."
  ConditionalConsistency:
    meaning: "Verify E[X]=E[E[X|Y]] (law of total expectation)."

# ---------- EQUATIONS ----------
equations:
  - "E[X] = \\int_{\\mathbb{R}} x f_X(x) dx"
  - "Var(X) = E[(X−E[X])²] = E[X²] − (E[X])²"
  - "Cov(X,Y) = E[XY] − E[X]E[Y]"
  - "ρ(X,Y) = \\frac{Cov(X,Y)}{σ_X σ_Y}"
  - "E[aX+b] = aE[X] + b"
  - "Var(aX+b) = a² Var(X)"
  - "Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)"
  - "E[g(X)] = \\int g(x) f_X(x) dx"
  - "E[X] = E[E[X|Y]], \\quad Var(X) = E[Var(X|Y)] + Var(E[X|Y])"
  - "LOTUS: E[g(X)] = \\int g(x) f_X(x) dx"

# ---------- NOTES ----------
notes:
  - "Expectation is the field’s balance point — the mean position of coherence weighted by probability."
  - "Variance measures the energy of fluctuation — how far the field deviates from its equilibrium centre."
  - "Linearity of expectation reflects the superposition principle of probability fields — averages combine without interference."
  - "Variance’s quadratic form reflects the stability of deviations — energy scales with square of distance from equilibrium."
  - "Covariance captures joint variation: how two distributions reinforce or counteract each other’s deviations."
  - "The law of total variance decomposes uncertainty into internal (within-group) and external (between-group) components — nested coherence levels."
  - "In Flow Algebra, expectation and variance describe the dynamic geometry of a probability field: the mean as the centroid of coherence, and variance as its field tension or dispersion energy."