File: Mathlib/Probability/Distributions/Gamma.lean

full_name: lintegral_Iic_eq_lintegral_Iio_add_Icc
code: /-- A Lebesgue Integral from -∞ to y can be expressed as the sum of one from -∞ to 0 and 0 to x -/
lemma lintegral_Iic_eq_lintegral_Iio_add_Icc {y z : ℝ} (f : ℝ → ℝ≥0∞) (hzy : z ≤ y) :
    ∫⁻ x in Iic y, f x = (∫⁻ x in Iio z, f x) + ∫⁻ x in Icc z y, f x 
goal: ∫⁻ (x : ℝ) in Set.Iic y, f x = (∫⁻ (x : ℝ) in Set.Iio z, f x) + ∫⁻ (x : ℝ) in Set.Icc z y, f x

full_name: ProbabilityTheory.gammaPDFReal
code: /-- The pdf of the gamma distribution depending on its scale and rate -/
noncomputable
def gammaPDFReal (a r x : ℝ) : ℝ 
goal: ℝ

full_name: ProbabilityTheory.gammaPDF
code: /-- The pdf of the gamma distribution, as a function valued in `ℝ≥0∞` -/
noncomputable
def gammaPDF (a r x : ℝ) : ℝ≥0∞ 
goal: ℝ≥0∞

full_name: ProbabilityTheory.gammaPDF_eq
code: lemma gammaPDF_eq (a r x : ℝ) :
    gammaPDF a r x = ENNReal.ofReal (if 0 ≤ x then
    r ^ a / (Gamma a) * x ^ (a-1) * exp (-(r * x)) else 0) 
goal: ProbabilityTheory.gammaPDF a r x =
  ENNReal.ofReal (if 0 ≤ x then r ^ a / Real.Gamma a * x ^ (a - 1) * rexp (-(r * x)) else 0)

full_name: ProbabilityTheory.gammaPDF_of_neg
code: lemma gammaPDF_of_neg {a r x : ℝ} (hx : x < 0) : gammaPDF a r x = 0 
goal: ProbabilityTheory.gammaPDF a r x = 0

full_name: ProbabilityTheory.gammaPDF_of_nonneg
code: lemma gammaPDF_of_nonneg {a r x : ℝ} (hx : 0 ≤ x) :
    gammaPDF a r x = ENNReal.ofReal (r ^ a / (Gamma a) * x ^ (a-1) * exp (-(r * x))) 
goal: ProbabilityTheory.gammaPDF a r x = ENNReal.ofReal (r ^ a / Real.Gamma a * x ^ (a - 1) * rexp (-(r * x)))

full_name: ProbabilityTheory.lintegral_gammaPDF_of_nonpos
code: /-- The Lebesgue integral of the gamma pdf over nonpositive reals equals 0 -/
lemma lintegral_gammaPDF_of_nonpos {x a r : ℝ} (hx : x ≤ 0) :
    ∫⁻ y in Iio x, gammaPDF a r y = 0 
goal: ∫⁻ (y : ℝ) in Set.Iio x, ProbabilityTheory.gammaPDF a r y = 0

full_name: ProbabilityTheory.measurable_gammaPDFReal
code: /-- The gamma pdf is measurable. -/
@[measurability]
lemma measurable_gammaPDFReal (a r : ℝ) : Measurable (gammaPDFReal a r) 
goal: Measurable (ProbabilityTheory.gammaPDFReal a r)

full_name: ProbabilityTheory.stronglyMeasurable_gammaPDFReal
code: /-- The gamma pdf is strongly measurable -/
@[measurability]
 lemma stronglyMeasurable_gammaPDFReal (a r : ℝ) :
     StronglyMeasurable (gammaPDFReal a r) 
goal: MeasureTheory.StronglyMeasurable (ProbabilityTheory.gammaPDFReal a r)

full_name: ProbabilityTheory.gammaPDFReal_pos
code: /-- The gamma pdf is positive for all positive reals -/
lemma gammaPDFReal_pos {x a r : ℝ} (ha : 0 < a) (hr : 0 < r) (hx : 0 < x) :
    0 < gammaPDFReal a r x 
goal: 0 < ProbabilityTheory.gammaPDFReal a r x

full_name: ProbabilityTheory.gammaPDFReal_nonneg
code: /-- The gamma pdf is nonnegative -/
lemma gammaPDFReal_nonneg {a r : ℝ} (ha : 0 < a) (hr : 0 < r) (x : ℝ) :
    0 ≤ gammaPDFReal a r x 
goal: 0 ≤ ProbabilityTheory.gammaPDFReal a r x

full_name: ProbabilityTheory.lintegral_gammaPDF_eq_one
code: /-- The pdf of the gamma distribution integrates to 1 -/
@[simp]
lemma lintegral_gammaPDF_eq_one {a r : ℝ} (ha : 0 < a) (hr : 0 < r) :
    ∫⁻ x, gammaPDF a r x = 1 
goal: ∫⁻ (x : ℝ), ProbabilityTheory.gammaPDF a r x = 1

full_name: ProbabilityTheory.gammaMeasure
code: /-- Measure defined by the gamma distribution -/
noncomputable
def gammaMeasure (a r : ℝ) : Measure ℝ 
goal: MeasureTheory.Measure ℝ

full_name: ProbabilityTheory.isProbabilityMeasureGamma
code: lemma isProbabilityMeasureGamma {a r : ℝ} (ha : 0 < a) (hr : 0 < r) :
    IsProbabilityMeasure (gammaMeasure a r) 
goal: MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.gammaMeasure a r)

full_name: ProbabilityTheory.gammaCDFReal
code: /-- CDF of the gamma distribution -/
noncomputable
def gammaCDFReal (a r : ℝ) : StieltjesFunction 
goal: StieltjesFunction

full_name: ProbabilityTheory.gammaCDFReal_eq_integral
code: lemma gammaCDFReal_eq_integral {a r : ℝ} (ha : 0 < a) (hr : 0 < r) (x : ℝ) :
    gammaCDFReal a r x = ∫ x in Iic x, gammaPDFReal a r x 
goal: ↑(ProbabilityTheory.gammaCDFReal a r) x = ∫ (x : ℝ) in Set.Iic x, ProbabilityTheory.gammaPDFReal a r x

full_name: ProbabilityTheory.gammaCDFReal_eq_lintegral
code: lemma gammaCDFReal_eq_lintegral {a r : ℝ} (ha : 0 < a) (hr : 0 < r) (x : ℝ) :
    gammaCDFReal a r x = ENNReal.toReal (∫⁻ x in Iic x, gammaPDF a r x) 
goal: ↑(ProbabilityTheory.gammaCDFReal a r) x = (∫⁻ (x : ℝ) in Set.Iic x, ProbabilityTheory.gammaPDF a r x).toReal