Theorems · Theorem · probability
ProbabilityTheory.complexMGF_gaussianReal
∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {p : MeasureTheory.Measure Ω} {μ : ℝ} {v : NNReal} {X : Ω → ℝ},
MeasureTheory.Measure.map X p = ProbabilityTheory.gaussianReal μ v →
∀ (z : ℂ), ProbabilityTheory.complexMGF X p z = Complex.exp (z * ↑μ + ↑↑v * z ^ 2 / 2)The complex moment-generating function of a random variable with Gaussian distribution
with mean μ and variance v is given by z ↦ exp (z * μ + v * z ^ 2 / 2).
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- 0 results in Mathlib
- Foundations
- Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites15
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
- Complexstatement and proof · cited by 5,565
- NNRealstatement and proof · cited by 4,310
- Complex.ofRealstatement and proof · cited by 1,654
- NNReal.toRealstatement and proof · cited by 1,260
- MeasureTheory.Measure.mapstatement and proof · cited by 858
- AEMeasurableproof · cited by 840
- Complex.expstatement and proof · cited by 612
- ProbabilityTheory.gaussianRealstatement and proof · cited by 77
- ProbabilityTheory.complexMGFstatement and proof · cited by 23
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