condVar_ae_eq_condExp_sq_sub_sq_condExp

English

Let X be a real-valued random variable and m a sub-sigma-algebra of the underlying probability space. Then the conditional variance of X given m equals the conditional expectation of X^2 given m minus the square of the conditional expectation of X given m, almost surely.

Русский

Пусть X — случайная величина с значениями в ℝ, а m — подσ-алгебра пространства, на котором определяется вероятность. Тогда условная дисперсия X по m равна условному математическому ожиданию X^2 по m минус квадрат условного ожидания X по m, vrijwel почти surely.

LaTeX

$$$\\ Var[X; \\mu | m] \\;=\\;ᵃᵉ_\\mu\\;\\big(\\mu[X^2|m] - (\\mu[X|m])^2\\big)$$$

Lean4

theorem condVar_ae_eq_condExp_sq_sub_sq_condExp (hm : m ≤ m₀) [IsFiniteMeasure μ] (hX : MemLp X 2 μ) :
    Var[X; μ | m] =ᵐ[μ] μ[X ^ 2|m] - μ[X|m] ^ 2 := by
  calc
    Var[X; μ | m]
    _ = μ[X ^ 2 - 2 * X * μ[X|m] + μ[X|m] ^ 2|m] := by rw [condVar, sub_sq]
    _ =ᵐ[μ] μ[X ^ 2|m] - 2 * μ[X|m] ^ 2 + μ[X|m] ^ 2 :=
      by
      have aux₀ : Integrable (X ^ 2) μ := hX.integrable_sq
      have aux₁ : Integrable (2 * X * μ[X|m]) μ := by
        rw [mul_assoc]
        exact (memLp_one_iff_integrable.1 <| hX.condExp.mul hX).const_mul _
      have aux₂ : Integrable (μ[X|m] ^ 2) μ := hX.condExp.integrable_sq
      filter_upwards [condExp_add (m := m) (aux₀.sub aux₁) aux₂, condExp_sub (m := m) aux₀ aux₁,
        condExp_mul_of_stronglyMeasurable_right stronglyMeasurable_condExp aux₁
          ((hX.integrable one_le_two).const_mul _),
        condExp_ofNat (m := m) 2 X] with ω hω₀ hω₁ hω₂ hω₃
      simp [hω₀, hω₁, hω₂, hω₃, condExp_of_stronglyMeasurable hm (stronglyMeasurable_condExp.pow _) aux₂]
      simp [mul_assoc, sq]
    _ = μ[X ^ 2|m] - μ[X|m] ^ 2 := by ring

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