expect_add_expect_comm — Mathlib

English

Swiftly, the average of a sum equals the sum of averages: E i∈s ∑ j∈t f(i,j) = ∑ j∈t E i∈s f(i,j).

Русский

Среднее от суммы равен сумме средних: E_i∈s ∑_j∈t f(i,j) = ∑_j∈t E_i∈s f(i,j).

LaTeX

$$$$ \\mathbb{E}_{i \\in s} \\sum_{j \\in t} f(i,j) = \\sum_{j \\in t} \\mathbb{E}_{i \\in s} f(i,j). $$$$

Lean4

theorem expect_add_expect_comm (f₁ f₂ g₁ g₂ : ι → M) :
    𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) = 𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i) := by
  simp_rw [expect_add_distrib, add_add_add_comm]

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