[Merged by Bors] - feat(Probability/Moments): add mgf_smul_left#19798
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[Merged by Bors] - feat(Probability/Moments): add mgf_smul_left#19798Vilin97 wants to merge 2 commits into
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PR summary 2e46ac1b16Import changes for modified filesNo significant changes to the import graph Import changes for all files
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| theorem cgf_neg : cgf (-X) μ t = cgf X μ (-t) := by simp_rw [cgf, mgf_neg] | ||
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| theorem mgf_smul_left (α : ℝ) : mgf (α • X) μ t = mgf X μ (t * α) := by | ||
| simp_rw [mgf, Pi.smul_apply, smul_eq_mul, mul_assoc] |
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| simp_rw [mgf, Pi.smul_apply, smul_eq_mul, mul_assoc] | |
| simp_rw [mgf, Pi.smul_apply, smul_eq_mul, mul_assoc] |
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Is there a good reason for reordering the variables? We have alpha to the left of t on one side, and t to the left of alpha on the other side.
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The reordering is an illusion. After an application of mgf, the goal is ∫ (x : Ω), rexp (t * (α • X) x) ∂μ = ∫ (x : Ω), rexp (t * α * X x) ∂μ. Having said that, I have now switched the statement to α * t with an extra application of mul_comm
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This seems fine.
maintainer merge
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🚀 Pull request has been placed on the maintainer queue by Ruben-VandeVelde. |
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Add `mgf_smul_left`. While there is a lemma [IndepFun.mgf_add](https://leanprover-community.github.io/mathlib4_docs/Mathlib/Probability/Moments.html#ProbabilityTheory.IndepFun.mgf_add) that describes the additive behavior of the moment generating function, there is currently no lemma describing the (scalar-)multiplicative behavior. This lemma is one of the ~4 supporting lemmas we have proved on our way to formalizing the central limit theorem. We will make PRs for other lemmas as well. Co-authored-by: Siyuan Ge <[email protected]>
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mgf_smul_left.While there is a lemma IndepFun.mgf_add that describes the additive behavior of the moment generating function, there is currently no lemma describing the (scalar-)multiplicative behavior.
This lemma is one of the ~4 supporting lemmas we have proved on our way to formalizing the central limit theorem. We will make PRs for other lemmas as well.
Co-authored-by: Siyuan Ge [email protected]