Theorems · Theorem · probability
ProbabilityTheory.integral_binomial
∀ {n : ℕ} {p : ↑unitInterval} {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [CompleteSpace E]
(f : ℕ → E),
∫ (x : ℕ), f x ∂ProbabilityTheory.binomial n p = ∑ k ≤ n, (↑(n.choose k) * ↑p ^ k * (1 - ↑p) ^ (n - k)) • f k- Cited by
- 1 results in Mathlib
- Foundations
- Depth 293 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- MeasureTheory.Measureproof · cited by 10,939
- Set.Elemstatement and proof · cited by 7,166
- Finset.sumstatement and proof · cited by 5,195
- CompleteSpacestatement and proof · cited by 2,532
- Finset.sum_congrproof · cited by 2,323
- MeasureTheory.integralstatement and proof · cited by 1,779
- ENNReal.ofRealproof · cited by 863
- unitIntervalstatement and proof · cited by 607
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.integral_map_cast_binomialproof · cited by 0