Theorems · Theorem · probability

ProbabilityTheory.exp_cgf

∀ {Ω : Type u_1} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : MeasureTheory.Measure Ω} {t : ℝ} [hμ : NeZero μ],
  MeasureTheory.Integrable (fun ω => Real.exp (t * X ω)) μ →
    Real.exp (ProbabilityTheory.cgf X μ t) = ProbabilityTheory.mgf X μ t

Defined in: Mathlib.Probability.Moments.Basic
Cited by: 8 results in Mathlib
Foundations: Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NeZero

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Cites9

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
MeasureTheory.Integrablestatement and proof · cited by 1,367
Real.expstatement and proof · cited by 871
ProbabilityTheory.mgfstatement and proof · cited by 113
Real.exp_logproof · cited by 58
ProbabilityTheory.cgfstatement · cited by 36
ProbabilityTheory.mgf_pos'proof · cited by 9

Cited by8

Results whose statement or proof uses this declaration.

ProbabilityTheory.mgf_le_of_mem_Icc_of_integral_eq_zeroproof · cited by 1
ProbabilityTheory.setIntegral_tilted_mul_eq_cgf'proof · cited by 0
ProbabilityTheory.tilted_mul_apply_cgfproof · cited by 0
ProbabilityTheory.tilted_mul_apply_cgf'proof · cited by 0
ProbabilityTheory.tilted_mul_apply_eq_ofReal_integral_cgfproof · cited by 0
ProbabilityTheory.tilted_mul_apply_eq_ofReal_integral_cgf'proof · cited by 0
ProbabilityTheory.integral_tilted_mul_eq_cgfproof · cited by 0
ProbabilityTheory.setIntegral_tilted_mul_eq_cgfproof · cited by 0