state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ (Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * (|c| * |c|⁻¹) =
(Real.sqrt ↑v)⁻¹ * (Real.sqrt 2)⁻¹ * (Real.sqrt π)⁻¹ * |c| * |c|⁻¹ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ring | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_2
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) = rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v))) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | congr 1 | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_2.e_x
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ -(c⁻¹ * x - μ) ^ 2 / (2 * ↑v) = -(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | field_simp | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_2.e_x
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ -(x - c * μ) ^ 2 / (c ^ 2 * (2 * ↑v)) = -(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | congr 1 | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
case refine_2.e_x.e_a
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ c ^ 2 * (2 * ↑v) = 2 * (c ^ 2 * ↑v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ring | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x := by
simp only [gaussianPdfReal._eq_1, gt_iff_lt, zero_lt_two, zero_le_mul_left, NNReal.zero_le_coe,
Real.sqrt_mul', one_div, mul_inv_rev, NN... | Mathlib.Probability.Distributions.Gaussian.131_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_inv_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c⁻¹ * x) = |c| * gaussianPdfReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ gaussianPdfReal μ v (c * x) =
|c⁻¹| * gaussianPdfReal (c⁻¹ * μ) ({ val := (c ^ 2)⁻¹, property := (_ : 0 ≤ (c ^ 2)⁻¹) } * v) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | conv_lhs => rw [← inv_inv c, gaussianPdfReal_inv_mul (inv_ne_zero hc)] | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x := by
| Mathlib.Probability.Distributions.Gaussian.150_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
| gaussianPdfReal μ v (c * x) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← inv_inv c, gaussianPdfReal_inv_mul (inv_ne_zero hc)] | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x := by
conv_lhs => | Mathlib.Probability.Distributions.Gaussian.150_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
| gaussianPdfReal μ v (c * x) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← inv_inv c, gaussianPdfReal_inv_mul (inv_ne_zero hc)] | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x := by
conv_lhs => | Mathlib.Probability.Distributions.Gaussian.150_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
| gaussianPdfReal μ v (c * x) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← inv_inv c, gaussianPdfReal_inv_mul (inv_ne_zero hc)] | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x := by
conv_lhs => | Mathlib.Probability.Distributions.Gaussian.150_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hc : c ≠ 0
x : ℝ
⊢ |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) ({ val := c⁻¹ ^ 2, property := (_ : 0 ≤ c⁻¹ ^ 2) } * v) x =
|c⁻¹| * gaussianPdfReal (c⁻¹ * μ) ({ val := (c ^ 2)⁻¹, property := (_ : 0 ≤ (c ^ 2)⁻¹) } * v) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x := by
conv_lhs => rw [← inv_inv c, gaussianPdfReal_inv_mul (inv_ne_zero hc)]
| Mathlib.Probability.Distributions.Gaussian.150_0.7VInOP4QlYS5vmc | lemma gaussianPdfReal_mul {μ : ℝ} {v : ℝ≥0} {c : ℝ} (hc : c ≠ 0) (x : ℝ) :
gaussianPdfReal μ v (c * x)
= |c⁻¹| * gaussianPdfReal (c⁻¹ * μ) (⟨(c^2)⁻¹, inv_nonneg.mpr (sq_nonneg _)⟩ * v) x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
⊢ gaussianPdf μ 0 = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ext | @[simp]
lemma gaussianPdf_zero_var (μ : ℝ) : gaussianPdf μ 0 = 0 := by
| Mathlib.Probability.Distributions.Gaussian.163_0.7VInOP4QlYS5vmc | @[simp]
lemma gaussianPdf_zero_var (μ : ℝ) : gaussianPdf μ 0 = 0 | Mathlib_Probability_Distributions_Gaussian |
case h
μ x✝ : ℝ
⊢ gaussianPdf μ 0 x✝ = OfNat.ofNat 0 x✝ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [gaussianPdf] | @[simp]
lemma gaussianPdf_zero_var (μ : ℝ) : gaussianPdf μ 0 = 0 := by
ext
| Mathlib.Probability.Distributions.Gaussian.163_0.7VInOP4QlYS5vmc | @[simp]
lemma gaussianPdf_zero_var (μ : ℝ) : gaussianPdf μ 0 = 0 | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
x : ℝ
⊢ 0 < gaussianPdf μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianPdf, ENNReal.ofReal_pos] | lemma gaussianPdf_pos (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : 0 < gaussianPdf μ v x := by
| Mathlib.Probability.Distributions.Gaussian.168_0.7VInOP4QlYS5vmc | lemma gaussianPdf_pos (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : 0 < gaussianPdf μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
x : ℝ
⊢ 0 < gaussianPdfReal μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianPdfReal_pos _ _ _ hv | lemma gaussianPdf_pos (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : 0 < gaussianPdf μ v x := by
rw [gaussianPdf, ENNReal.ofReal_pos]
| Mathlib.Probability.Distributions.Gaussian.168_0.7VInOP4QlYS5vmc | lemma gaussianPdf_pos (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (x : ℝ) : 0 < gaussianPdf μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
⊢ ↑↑(gaussianReal μ v) Set.univ = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases h : v = 0 | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ := by | Mathlib.Probability.Distributions.Gaussian.196_0.7VInOP4QlYS5vmc | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
h : v = 0
⊢ ↑↑(gaussianReal μ v) Set.univ = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [gaussianReal_of_var_ne_zero, h] | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ := by by_cases h : v = 0 <;> | Mathlib.Probability.Distributions.Gaussian.196_0.7VInOP4QlYS5vmc | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
h : ¬v = 0
⊢ ↑↑(gaussianReal μ v) Set.univ = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [gaussianReal_of_var_ne_zero, h] | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ := by by_cases h : v = 0 <;> | Mathlib.Probability.Distributions.Gaussian.196_0.7VInOP4QlYS5vmc | instance instIsProbabilityMeasureGaussianReal (μ : ℝ) (v : ℝ≥0) :
IsProbabilityMeasure (gaussianReal μ v) where
measure_univ | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ↑↑(gaussianReal μ v) s = ∫⁻ (x : ℝ) in s, gaussianPdf μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv, withDensity_apply' _ s] | lemma gaussianReal_apply (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ∫⁻ x in s, gaussianPdf μ v x := by
| Mathlib.Probability.Distributions.Gaussian.200_0.7VInOP4QlYS5vmc | lemma gaussianReal_apply (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ∫⁻ x in s, gaussianPdf μ v x | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ↑↑(gaussianReal μ v) s = ENNReal.ofReal (∫ (x : ℝ) in s, gaussianPdfReal μ v x) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_apply _ hv s, ofReal_integral_eq_lintegral_ofReal] | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) := by
| Mathlib.Probability.Distributions.Gaussian.204_0.7VInOP4QlYS5vmc | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ ∫⁻ (x : ℝ) in s, gaussianPdf μ v x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal (gaussianPdfReal μ v x) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rfl | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) := by
rw [gaussianReal_apply _ hv s, ofReal_integral_eq_lintegral_ofReal]
· | Mathlib.Probability.Distributions.Gaussian.204_0.7VInOP4QlYS5vmc | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) | Mathlib_Probability_Distributions_Gaussian |
case hfi
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ Integrable fun x => gaussianPdfReal μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact (integrable_gaussianPdfReal _ _).restrict | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) := by
rw [gaussianReal_apply _ hv s, ofReal_integral_eq_lintegral_ofReal]
· rfl
· | Mathlib.Probability.Distributions.Gaussian.204_0.7VInOP4QlYS5vmc | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) | Mathlib_Probability_Distributions_Gaussian |
case f_nn
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
s : Set ℝ
⊢ 0 ≤ᵐ[Measure.restrict ℙ s] fun x => gaussianPdfReal μ v x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact ae_of_all _ (gaussianPdfReal_nonneg _ _) | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) := by
rw [gaussianReal_apply _ hv s, ofReal_integral_eq_lintegral_ofReal]
· rfl
· exact (integrable_gaussianPdfReal _ _).restrict
· | Mathlib.Probability.Distributions.Gaussian.204_0.7VInOP4QlYS5vmc | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPdfReal μ v x) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ gaussianReal μ v ≪ ℙ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv] | lemma gaussianReal_absolutelyContinuous (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
gaussianReal μ v ≪ volume := by
| Mathlib.Probability.Distributions.Gaussian.211_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
gaussianReal μ v ≪ volume | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ Measure.withDensity ℙ (gaussianPdf μ v) ≪ ℙ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact withDensity_absolutelyContinuous _ _ | lemma gaussianReal_absolutelyContinuous (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
gaussianReal μ v ≪ volume := by
rw [gaussianReal_of_var_ne_zero _ hv]
| Mathlib.Probability.Distributions.Gaussian.211_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
gaussianReal μ v ≪ volume | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ ≪ gaussianReal μ v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv] | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v := by
| Mathlib.Probability.Distributions.Gaussian.216_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ℙ ≪ Measure.withDensity ℙ (gaussianPdf μ v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | refine withDensity_absolutelyContinuous' ?_ ?_ ?_ | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v := by
rw [gaussianReal_of_var_ne_zero _ hv]
| Mathlib.Probability.Distributions.Gaussian.216_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v | Mathlib_Probability_Distributions_Gaussian |
case refine_1
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ AEMeasurable (gaussianPdf μ v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact (measurable_gaussianPdf _ _).aemeasurable | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v := by
rw [gaussianReal_of_var_ne_zero _ hv]
refine withDensity_absolutelyContinuous' ?_ ?_ ?_
· | Mathlib.Probability.Distributions.Gaussian.216_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v | Mathlib_Probability_Distributions_Gaussian |
case refine_2
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ∀ᵐ (x : ℝ), gaussianPdf μ v x ≠ 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact ae_of_all _ (fun _ ↦ (gaussianPdf_pos _ hv _).ne') | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v := by
rw [gaussianReal_of_var_ne_zero _ hv]
refine withDensity_absolutelyContinuous' ?_ ?_ ?_
· exact (measurable_gaussianPdf _ _).aemeasurable
· | Mathlib.Probability.Distributions.Gaussian.216_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v | Mathlib_Probability_Distributions_Gaussian |
case refine_3
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
⊢ ∀ᵐ (x : ℝ), gaussianPdf μ v x ≠ ⊤ | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact ae_of_all _ (fun _ ↦ ENNReal.ofReal_ne_top) | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v := by
rw [gaussianReal_of_var_ne_zero _ hv]
refine withDensity_absolutelyContinuous' ?_ ?_ ?_
· exact (measurable_gaussianPdf _ _).aemeasurable
· exact ae_of_all _ (fun _ ↦ (gaussianPdf_pos _ hv _).ne')
· | Mathlib.Probability.Distributions.Gaussian.216_0.7VInOP4QlYS5vmc | lemma gaussianReal_absolutelyContinuous' (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
volume ≪ gaussianReal μ v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
⊢ ∂gaussianReal μ v/∂ℙ =ᵐ[ℙ] gaussianPdf μ v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases hv : v = 0 | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
| Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
hv : v = 0
⊢ ∂gaussianReal μ v/∂ℙ =ᵐ[ℙ] gaussianPdf μ v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [hv, gaussianReal_zero_var, gaussianPdf_zero_var] | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
by_cases hv : v = 0
· | Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
hv : v = 0
⊢ ∂Measure.dirac μ/∂ℙ =ᵐ[ℙ] 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | refine (Measure.eq_rnDeriv measurable_zero (mutuallySingular_dirac μ volume) ?_).symm | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
by_cases hv : v = 0
· simp only [hv, gaussianReal_zero_var, gaussianPdf_zero_var]
| Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
hv : v = 0
⊢ Measure.dirac μ = Measure.dirac μ + Measure.withDensity ℙ 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [withDensity_zero, add_zero] | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
by_cases hv : v = 0
· simp only [hv, gaussianReal_zero_var, gaussianPdf_zero_var]
refine (Measure.eq_rnDeriv measurable_zero (mutuallySingular_dirac μ volume) ?_).symm
| Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
⊢ ∂gaussianReal μ v/∂ℙ =ᵐ[ℙ] gaussianPdf μ v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv] | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
by_cases hv : v = 0
· simp only [hv, gaussianReal_zero_var, gaussianPdf_zero_var]
refine (Measure.eq_rnDeriv measurable_zero (mutuallySingular_dirac μ volume) ?_).symm
rw [withDensity_zero, add_zero]
·... | Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
hv : ¬v = 0
⊢ ∂Measure.withDensity ℙ (gaussianPdf μ v)/∂ℙ =ᵐ[ℙ] gaussianPdf μ v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact Measure.rnDeriv_withDensity _ (measurable_gaussianPdf μ v) | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v := by
by_cases hv : v = 0
· simp only [hv, gaussianReal_zero_var, gaussianPdf_zero_var]
refine (Measure.eq_rnDeriv measurable_zero (mutuallySingular_dirac μ volume) ?_).symm
rw [withDensity_zero, add_zero]
·... | Mathlib.Probability.Distributions.Gaussian.224_0.7VInOP4QlYS5vmc | lemma rnDeriv_gaussianReal (μ : ℝ) (v : ℝ≥0) :
∂(gaussianReal μ v)/∂volume =ₐₛ gaussianPdf μ v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ → ℝ
hf : MeasurableEmbedding f
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt f (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ ↑↑(Measure.comap f (gaussianReal μ v)) s = ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPdfReal μ v (f x)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv, gaussianPdf_def] | lemma _root_.MeasurableEmbedding.gaussianReal_comap_apply (hv : v ≠ 0)
{f : ℝ → ℝ} (hf : MeasurableEmbedding f)
{f' : ℝ → ℝ} (h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).comap f s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) := by
| Mathlib.Probability.Distributions.Gaussian.237_0.7VInOP4QlYS5vmc | lemma _root_.MeasurableEmbedding.gaussianReal_comap_apply (hv : v ≠ 0)
{f : ℝ → ℝ} (hf : MeasurableEmbedding f)
{f' : ℝ → ℝ} (h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).comap f s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ → ℝ
hf : MeasurableEmbedding f
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt f (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ ↑↑(Measure.comap f (Measure.withDensity ℙ fun x => ENNReal.ofReal (gaussianPdfReal μ v x))) s =
ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPdfReal μ v (f x)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact hf.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul' hs h_deriv
(ae_of_all _ (gaussianPdfReal_nonneg _ _)) (integrable_gaussianPdfReal _ _) | lemma _root_.MeasurableEmbedding.gaussianReal_comap_apply (hv : v ≠ 0)
{f : ℝ → ℝ} (hf : MeasurableEmbedding f)
{f' : ℝ → ℝ} (h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).comap f s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) := by
... | Mathlib.Probability.Distributions.Gaussian.237_0.7VInOP4QlYS5vmc | lemma _root_.MeasurableEmbedding.gaussianReal_comap_apply (hv : v ≠ 0)
{f : ℝ → ℝ} (hf : MeasurableEmbedding f)
{f' : ℝ → ℝ} (h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).comap f s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ ≃ᵐ ℝ
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ ↑↑(Measure.map (⇑f.symm) (gaussianReal μ v)) s = ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPdfReal μ v (f x)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_of_var_ne_zero _ hv, gaussianPdf_def] | lemma _root_.MeasurableEquiv.gaussianReal_map_symm_apply (hv : v ≠ 0) (f : ℝ ≃ᵐ ℝ) {f' : ℝ → ℝ}
(h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).map f.symm s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) := by
| Mathlib.Probability.Distributions.Gaussian.246_0.7VInOP4QlYS5vmc | lemma _root_.MeasurableEquiv.gaussianReal_map_symm_apply (hv : v ≠ 0) (f : ℝ ≃ᵐ ℝ) {f' : ℝ → ℝ}
(h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).map f.symm s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
hv : v ≠ 0
f : ℝ ≃ᵐ ℝ
f' : ℝ → ℝ
h_deriv : ∀ (x : ℝ), HasDerivAt (⇑f) (f' x) x
s : Set ℝ
hs : MeasurableSet s
⊢ ↑↑(Measure.map (⇑f.symm) (Measure.withDensity ℙ fun x => ENNReal.ofReal (gaussianPdfReal μ v x))) s =
ENNReal.ofReal (∫ (x : ℝ) in s, |f' x| * gaussianPdfReal μ v (f x)) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact f.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul' hs h_deriv
(ae_of_all _ (gaussianPdfReal_nonneg _ _)) (integrable_gaussianPdfReal _ _) | lemma _root_.MeasurableEquiv.gaussianReal_map_symm_apply (hv : v ≠ 0) (f : ℝ ≃ᵐ ℝ) {f' : ℝ → ℝ}
(h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).map f.symm s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) := by
rw [gaussianReal_of_var_ne_zer... | Mathlib.Probability.Distributions.Gaussian.246_0.7VInOP4QlYS5vmc | lemma _root_.MeasurableEquiv.gaussianReal_map_symm_apply (hv : v ≠ 0) (f : ℝ ≃ᵐ ℝ) {f' : ℝ → ℝ}
(h_deriv : ∀ x, HasDerivAt f (f' x) x) {s : Set ℝ} (hs : MeasurableSet s) :
(gaussianReal μ v).map f.symm s
= ENNReal.ofReal (∫ x in s, |f' x| * gaussianPdfReal μ v (f x)) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
y : ℝ
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases hv : v = 0 | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
| Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
y : ℝ
hv : v = 0
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [hv, ne_eq, not_true, gaussianReal_zero_var] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
y : ℝ
hv : v = 0
⊢ Measure.map (fun x => x + y) (Measure.dirac μ) = Measure.dirac (μ + y) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact Measure.map_dirac (measurable_id'.add_const _) _ | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
| Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | let e : ℝ ≃ᵐ ℝ := (Homeomorph.addRight y).symm.toMeasurableEquiv | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have he' : ∀ x, HasDerivAt e ((fun _ ↦ 1) x) x := fun _ ↦ (hasDerivAt_id _).sub_const y | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | change (gaussianReal μ v).map e.symm = gaussianReal (μ + y) v | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
⊢ Measure.map (⇑e.symm) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ext s' hs' | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ↑↑(Measure.map (⇑e.symm) (gaussianReal μ v)) s' = ↑↑(gaussianReal (μ + y) v) s' | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [MeasurableEquiv.gaussianReal_map_symm_apply hv e he' hs'] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |(fun x => 1) x| * gaussianPdfReal μ v (e x)) = ↑↑(gaussianReal (μ + y)... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [abs_neg, abs_one, MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, one_mul, ne_eq] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s',
gaussianPdfReal μ v ((Homeomorph.toMeasurableEquiv (Homeo... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_apply_eq_integral _ hv s'] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
y : ℝ
hv : ¬v = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.addRight y))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => 1) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s',
gaussianPdfReal μ v ((Homeomorph.toMeasurableEquiv (Homeo... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [gaussianPdfReal_sub _ y, Homeomorph.addRight, ← sub_eq_add_neg] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v := by
by_cases hv : v = 0
· simp only [hv, ne_eq, not_true, gaussianReal_zero_var]
exact Measure.map_dirac (measurable_id'.add... | Mathlib.Probability.Distributions.Gaussian.254_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_add_const (y : ℝ) :
(gaussianReal μ v).map (· + y) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
y : ℝ
⊢ Measure.map (fun x => y + x) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp_rw [add_comm y] | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_const_add (y : ℝ) :
(gaussianReal μ v).map (y + ·) = gaussianReal (μ + y) v := by
| Mathlib.Probability.Distributions.Gaussian.269_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_const_add (y : ℝ) :
(gaussianReal μ v).map (y + ·) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
y : ℝ
⊢ Measure.map (fun x => x + y) (gaussianReal μ v) = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianReal_map_add_const y | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_const_add (y : ℝ) :
(gaussianReal μ v).map (y + ·) = gaussianReal (μ + y) v := by
simp_rw [add_comm y]
| Mathlib.Probability.Distributions.Gaussian.269_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by addition of a constant is a Gaussian. -/
lemma gaussianReal_map_const_add (y : ℝ) :
(gaussianReal μ v).map (y + ·) = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases hv : v = 0 | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
| Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : v = 0
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : v = 0
⊢ Measure.map (fun x => c * x) (Measure.dirac μ) = Measure.dirac (c * μ) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact Measure.map_dirac (measurable_id'.const_mul c) μ | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
| Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | by_cases hc : c = 0 | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [hc, zero_mul, ne_eq, abs_zero, mul_eq_zero] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ Measure.map (fun x => 0) (gaussianReal μ v) =
gaussianReal 0 ({ val := 0 ^ 2, property := (_ : (fun r => 0 ≤ r) (0 ^ 2)) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [Measure.map_const] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ ↑↑(gaussianReal μ v) Set.univ • Measure.dirac 0 =
gaussianReal 0 ({ val := 0 ^ 2, property := (_ : (fun r => 0 ≤ r) (0 ^ 2)) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [ne_eq, measure_univ, one_smul, mul_eq_zero] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case pos
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ Measure.dirac 0 = gaussianReal 0 ({ val := 0 ^ 2, property := (_ : (fun r => 0 ≤ r) (0 ^ 2)) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | convert (gaussianReal_zero_var 0).symm | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case h.e'_3.h.e'_2
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ { val := 0 ^ 2, property := (_ : (fun r => 0 ≤ r) (0 ^ 2)) } * v = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [ne_eq, zero_pow', mul_eq_zero, hv, or_false, not_false_eq_true] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case h.e'_3.h.e'_2
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : c = 0
⊢ { val := 0, property := (_ : (fun r => 0 ≤ r) 0) } = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rfl | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | let e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have he' : ∀ x, HasDerivAt e ((fun _ ↦ c⁻¹) x) x := by
suffices ∀ x, HasDerivAt (fun x => c⁻¹ * x) (c⁻¹ * 1) x by rwa [mul_one] at this
exact fun _ ↦ HasDerivAt.const_mul _ (hasDerivAt_id _) | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
⊢ ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | suffices ∀ x, HasDerivAt (fun x => c⁻¹ * x) (c⁻¹ * 1) x by rwa [mul_one] at this | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
this : ∀ (x : ℝ), HasDerivAt (fun x => c⁻¹ * x) (c⁻¹ * 1) x
⊢ ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rwa [mul_one] at this | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
⊢ ∀ (x : ℝ), HasDerivAt (fun x => c⁻¹ * x) (c⁻¹ * 1) x | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact fun _ ↦ HasDerivAt.const_mul _ (hasDerivAt_id _) | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } *... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | change (gaussianReal μ v).map e.symm = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
⊢ Measure.map (⇑e.symm) (gaussianReal μ v) = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | ext s' hs' | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ↑↑(Measure.map (⇑e.symm) (gaussianReal μ v)) s' =
↑↑(gaussianReal (c * μ) ({ val :=... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [MeasurableEquiv.gaussianReal_map_symm_apply hv e he' hs'] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal (∫ (x : ℝ) in s', |(fun x => c⁻¹) x| * gaussianPdfReal μ v (e x)) =
... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [MeasurableEquiv.coe_mk, Equiv.coe_fn_mk, ne_eq, mul_eq_zero] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s',
|c⁻¹| * gaussianPdfReal μ v ((Homeomorph... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [gaussianReal_apply_eq_integral _ _ s'] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s',
|c⁻¹| * gaussianPdfReal μ v ((Homeomorph... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | swap | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ { val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v ≠ 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [ne_eq, mul_eq_zero, hv, or_false] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ¬{ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } = 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← NNReal.coe_eq] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ¬↑{ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } = ↑0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp [hc] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s',
|c⁻¹| * gaussianPdfReal μ v ((Homeomorph... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp only [Homeomorph.mulLeft₀, Equiv.toFun_as_coe, Equiv.mulLeft₀_apply, Equiv.invFun_as_coe,
Equiv.mulLeft₀_symm_apply, Homeomorph.toMeasurableEquiv_coe, Homeomorph.homeomorph_mk_coe_symm,
Equiv.coe_fn_symm_mk, gaussianPdfReal_inv_mul hc] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
⊢ ENNReal.ofReal
(∫ (x : ℝ) in s', |c⁻¹| * (|c| * gaussianPdfReal (c * μ) ({ val :=... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | congr with x | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h.e_r.e_f.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
⊢ |c⁻¹| * (|c| * gaussianPdfReal (c * μ) ({ val := c ^ 2, property := (_ ... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | suffices |c⁻¹| * |c| = 1 by rw [← mul_assoc, this, one_mul] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
this : |c⁻¹| * |c| = 1
⊢ |c⁻¹| * (|c| * gaussianPdfReal (c * μ) ({ val := c ^ 2, property := (... | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← mul_assoc, this, one_mul] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h.e_r.e_f.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
⊢ |c⁻¹| * |c| = 1 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [abs_inv, inv_mul_cancel] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
case neg.h.e_r.e_f.h
μ : ℝ
v : ℝ≥0
c : ℝ
hv : ¬v = 0
hc : ¬c = 0
e : ℝ ≃ᵐ ℝ := Homeomorph.toMeasurableEquiv (Homeomorph.symm (Homeomorph.mulLeft₀ c hc))
he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x
s' : Set ℝ
hs' : MeasurableSet s'
x : ℝ
⊢ |c| ≠ 0 | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rwa [ne_eq, abs_eq_zero] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
by_cases hv : v = 0
· simp only [hv, mul_zero, ne_eq, not_true, gaussianReal_zero_var]
exac... | Mathlib.Probability.Distributions.Gaussian.275_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_const_mul (c : ℝ) :
(gaussianReal μ v).map (c * ·) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
⊢ Measure.map (fun x => x * c) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp_rw [mul_comm _ c] | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_mul_const (c : ℝ) :
(gaussianReal μ v).map (· * c) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
| Mathlib.Probability.Distributions.Gaussian.309_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_mul_const (c : ℝ) :
(gaussianReal μ v).map (· * c) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
c : ℝ
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianReal_map_const_mul c | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_mul_const (c : ℝ) :
(gaussianReal μ v).map (· * c) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) := by
simp_rw [mul_comm _ c]
| Mathlib.Probability.Distributions.Gaussian.309_0.7VInOP4QlYS5vmc | /-- The map of a Gaussian distribution by multiplication by a constant is a Gaussian. -/
lemma gaussianReal_map_mul_const (c : ℝ) :
(gaussianReal μ v).map (· * c) = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
⊢ Measure.map (fun ω => X ω + y) ℙ = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have hXm : AEMeasurable X := aemeasurable_of_map_neZero (by rw [hX]; infer_instance) | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v := by
| Mathlib.Probability.Distributions.Gaussian.317_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
⊢ NeZero (Measure.map X ℙ) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [hX] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v := by
ha... | Mathlib.Probability.Distributions.Gaussian.317_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
⊢ NeZero (gaussianReal μ v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | infer_instance | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v := by
ha... | Mathlib.Probability.Distributions.Gaussian.317_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
hXm : AEMeasurable X
⊢ Measure.map (fun ω => X ω + y) ℙ = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | change Measure.map ((fun ω ↦ ω + y) ∘ X) ℙ = gaussianReal (μ + y) v | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v := by
ha... | Mathlib.Probability.Distributions.Gaussian.317_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
hXm : AEMeasurable X
⊢ Measure.map ((fun ω => ω + y) ∘ X) ℙ = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← AEMeasurable.map_map_of_aemeasurable (measurable_id'.add_const _).aemeasurable hXm, hX,
gaussianReal_map_add_const y] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v := by
ha... | Mathlib.Probability.Distributions.Gaussian.317_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X + y`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_add_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ X ω + y) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
⊢ Measure.map (fun ω => y + X ω) ℙ = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp_rw [add_comm y] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `y + X`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_const_add {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ y + X ω) ℙ = gaussianReal (μ + y) v := by
| Mathlib.Probability.Distributions.Gaussian.326_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `y + X`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_const_add {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ y + X ω) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
y : ℝ
⊢ Measure.map (fun ω => X ω + y) ℙ = gaussianReal (μ + y) v | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianReal_add_const hX y | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `y + X`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_const_add {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ y + X ω) ℙ = gaussianReal (μ + y) v := by
si... | Mathlib.Probability.Distributions.Gaussian.326_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `y + X`
has Gaussian law with mean `μ + y` and variance `v`. -/
lemma gaussianReal_const_add {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (y : ℝ) :
Measure.map (fun ω ↦ y + X ω) ℙ = gaussianReal (μ + y) v | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ Measure.map (fun ω => c * X ω) ℙ = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | have hXm : AEMeasurable X := aemeasurable_of_map_neZero (by rw [hX]; infer_instance) | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ NeZero (Measure.map X ℙ) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [hX] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ NeZero (gaussianReal μ v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | infer_instance | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
hXm : AEMeasurable X
⊢ Measure.map (fun ω => c * X ω) ℙ = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | change Measure.map ((fun ω ↦ c * ω) ∘ X) ℙ = gaussianReal (c * μ) (⟨c^2, sq_nonneg _⟩ * v) | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
hXm : AEMeasurable X
⊢ Measure.map ((fun ω => c * ω) ∘ X) ℙ = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | rw [← AEMeasurable.map_map_of_aemeasurable (measurable_id'.const_mul c).aemeasurable hXm, hX] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
hXm : AEMeasurable X
⊢ Measure.map (fun x => c * x) (gaussianReal μ v) =
gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianReal_map_const_mul c | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.333_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `c * X`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_const_mul {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ c * X ω) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ Measure.map (fun ω => X ω * c) ℙ = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | simp_rw [mul_comm _ c] | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X * c`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_mul_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ X ω * c) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.342_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X * c`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_mul_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ X ω * c) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
μ : ℝ
v : ℝ≥0
Ω : Type
inst✝ : MeasureSpace Ω
X : Ω → ℝ
hX : Measure.map X ℙ = gaussianReal μ v
c : ℝ
⊢ Measure.map (fun ω => c * X ω) ℙ = gaussianReal (c * μ) ({ val := c ^ 2, property := (_ : 0 ≤ c ^ 2) } * v) | /-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian
import Mathlib.Probability.Notation
/-!
# Gaussian distributions over ℝ
We defin... | exact gaussianReal_const_mul hX c | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X * c`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_mul_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ X ω * c) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib.Probability.Distributions.Gaussian.342_0.7VInOP4QlYS5vmc | /-- If `X` is a real random variable with Gaussian law with mean `μ` and variance `v`, then `X * c`
has Gaussian law with mean `c * μ` and variance `c^2 * v`. -/
lemma gaussianReal_mul_const {X : Ω → ℝ} (hX : Measure.map X ℙ = gaussianReal μ v) (c : ℝ) :
Measure.map (fun ω ↦ X ω * c) ℙ = gaussianReal (c * μ) (⟨c^2,... | Mathlib_Probability_Distributions_Gaussian |
α✝ : Type ?u.229
β✝ : Type ?u.232
γ✝ : Type ?u.235
f✝ : α✝ → β✝ → γ✝
a✝ : Option α✝
b✝ : Option β✝
c : Option γ✝
α β γ : Type u
f : α → β → γ
a : Option α
b : Option β
⊢ map₂ f a b = Seq.seq (f <$> a) fun x => b | /-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Option.Basic
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
/-!
# Binary map of opti... | cases a | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b :=
by | Mathlib.Data.Option.NAry.46_0.qAUZsy5CifjOLJ4 | /-- `Option.map₂` in terms of monadic operations. Note that this can't be taken as the definition
because of the lack of universe polymorphism. -/
theorem map₂_def {α β γ : Type u} (f : α → β → γ) (a : Option α) (b : Option β) :
map₂ f a b = f <$> a <*> b | Mathlib_Data_Option_NAry |
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