Theorems · Theorem · probability
ProbabilityTheory.integral_condVar_add_variance_condExp
∀ {Ω : Type u_1} {m₀ m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω},
m ≤ m₀ →
∀ [MeasureTheory.IsProbabilityMeasure μ],
MeasureTheory.MemLp X 2 μ →
∫ (x : Ω), ProbabilityTheory.condVar m X μ x ∂μ + ProbabilityTheory.variance μ[X | m] μ =
ProbabilityTheory.variance X μLaw of total variance
- Defined in
- Mathlib.Probability.CondVar
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 312 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement · cited by 9,879
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasureTheory.MemLpstatement and proof · cited by 457
- MeasureTheory.IsProbabilityMeasurestatement and proof · cited by 392
- MeasureTheory.condExpstatement and proof · cited by 234
- MeasureTheory.integral_congr_aeproof · cited by 110
- ProbabilityTheory.variancestatement and proof · cited by 104
- one_le_twoproof · cited by 44
- MeasureTheory.integrable_condExpproof · cited by 28
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