Theorems · Theorem · probability
ProbabilityTheory.variance_eq_sub
∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ]
{X : Ω → ℝ},
MeasureTheory.MemLp X 2 μ → ProbabilityTheory.variance X μ = ∫ (x : Ω), (X ^ 2) x ∂μ - (∫ (x : Ω), X x ∂μ) ^ 2- Defined in
- Mathlib.Probability.Moments.Variance
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- pow_twoproof · cited by 150
- ProbabilityTheory.variancestatement · cited by 104
- ProbabilityTheory.covariance_selfproof · cited by 11
- MeasureTheory.MemLp.aemeasurableproof · cited by 10
- ProbabilityTheory.covariance_eq_subproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- ProbabilityTheory.variance_le_expectation_sqproof · cited by 1
- ProbabilityTheory.variance_le_sub_mul_subproof · cited by 1
- ProbabilityTheory.integral_condVar_add_variance_condExpproof · cited by 0
- ProbabilityTheory.evariance_def'proof · cited by 0