geometricPMFRealSum — Mathlib

English

If X and Y are independent with Gaussian maps X ∼ GaussianReal m1 v1 and Y ∼ GaussianReal m2 v2, then X+Y ∼ GaussianReal (m1+m2) (v1+v2).

Русский

Если X и Y независимы и X ∼ GaussianReal(m1,v1), Y ∼ GaussianReal(m2,v2), то X+Y ∼ GaussianReal(m1+m2, v1+v2).

LaTeX

$$$\text{IndepFun } X Y P \rightarrow P\!\maxmap X = \mathcal{N}(m_1,v_1) \rightarrow P\!\maxmap Y = \mathcal{N}(m_2,v_2) \rightarrow P\!\maxmap (X+Y) = \mathcal{N}(m_1+m_2, v_1+v_2).$$$

Lean4

theorem geometricPMFRealSum (hp_pos : 0 < p) (hp_le_one : p ≤ 1) : HasSum (fun n ↦ geometricPMFReal p n) 1 :=
  by
  unfold geometricPMFReal
  have := hasSum_geometric_of_lt_one (sub_nonneg.mpr hp_le_one) (sub_lt_self 1 hp_pos)
  apply(hasSum_mul_right_iff (hp_pos.ne')).mpr at this
  simp only [sub_sub_cancel] at this
  rw [inv_mul_eq_div, div_self hp_pos.ne'] at this
  exact this

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