Theorems · Theorem · probability

ProbabilityTheory.mgf_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 →
    ∀ (t : ℝ), ProbabilityTheory.mgf X p t = Real.exp (μ * t + ↑v * t ^ 2 / 2)

The moment-generating function of a random variable with Gaussian distribution with mean μ and variance v is given by t ↦ exp (μ * t + v * t ^ 2 / 2).

Defined in

Mathlib.Probability.Distributions.Gaussian.Real

Cited by

3 results in Mathlib

Foundations

Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound