Theorems · Theorem · probability
ProbabilityTheory.charFun_gaussianReal
∀ {μ : ℝ} {v : NNReal} (t : ℝ),
MeasureTheory.charFun (ProbabilityTheory.gaussianReal μ v) t = Complex.exp (↑t * ↑μ * Complex.I - ↑↑v * ↑t ^ 2 / 2)The characteristic function of a Gaussian distribution with mean μ and variance v
is given by t ↦ exp (t * μ - v * t ^ 2 / 2).
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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
- Complexstatement and proof · cited by 5,565
- NNRealstatement and proof · cited by 4,310
- mul_oneproof · cited by 3,885
- add_zeroproof · cited by 2,707
- Complex.ofRealstatement and proof · cited by 1,654
- NNReal.toRealstatement and proof · cited by 1,260
- sub_eq_add_negproof · cited by 1,023
- pow_oneproof · cited by 894
- Complex.Istatement and proof · cited by 866
- Complex.expstatement and proof · cited by 612
- mul_negproof · cited by 590
Cited by7
Results whose statement or proof uses this declaration.
- ProbabilityTheory.IsGaussian.charFunDual_eqproof · cited by 9
- ProbabilityTheory.isGaussian_iff_gaussian_charFunDualproof · cited by 4
- ProbabilityTheory.charFun_stdGaussianproof · cited by 2
- ProbabilityTheory.isGaussian_iff_charFunDual_eqproof · cited by 2
- ProbabilityTheory.gaussianReal_conv_gaussianRealproof · cited by 1
- ProbabilityTheory.map_pi_eq_stdGaussianproof · cited by 1
- ProbabilityTheory.tendstoInDistribution_inv_sqrt_mul_sumproof · cited by 0