File: Mathlib/Probability/Distributions/Exponential.lean

full_name: ProbabilityTheory.exponentialPDFReal
code: /-- The pdf of the exponential distribution depending on its rate -/
noncomputable
def exponentialPDFReal (r x : ℝ) : ℝ 
goal: ℝ

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

full_name: ProbabilityTheory.exponentialPDF_eq
code: lemma exponentialPDF_eq (r x : ℝ) :
    exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * exp (-(r * x)) else 0) 
goal: ProbabilityTheory.exponentialPDF r x = ENNReal.ofReal (if 0 ≤ x then r * rexp (-(r * x)) else 0)

full_name: ProbabilityTheory.exponentialPDF_of_neg
code: lemma exponentialPDF_of_neg {r x : ℝ} (hx : x < 0) : exponentialPDF r x = 0 
goal: ProbabilityTheory.exponentialPDF r x = 0

full_name: ProbabilityTheory.exponentialPDF_of_nonneg
code: lemma exponentialPDF_of_nonneg {r x : ℝ} (hx : 0 ≤ x) :
    exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x))) 
goal: ProbabilityTheory.exponentialPDF r x = ENNReal.ofReal (r * rexp (-(r * x)))

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

full_name: ProbabilityTheory.measurable_exponentialPDFReal
code: /-- The exponential pdf is measurable. -/
@[measurability]
lemma measurable_exponentialPDFReal (r : ℝ) : Measurable (exponentialPDFReal r) 
goal: Measurable (ProbabilityTheory.exponentialPDFReal r)

full_name: ProbabilityTheory.stronglyMeasurable_exponentialPDFReal
code: @[measurability]
 lemma stronglyMeasurable_exponentialPDFReal (r : ℝ) :
     StronglyMeasurable (exponentialPDFReal r) 
goal: MeasureTheory.StronglyMeasurable (ProbabilityTheory.exponentialPDFReal r)

full_name: ProbabilityTheory.exponentialPDFReal_pos
code: /-- The exponential pdf is positive for all positive reals -/
lemma exponentialPDFReal_pos {x r : ℝ} (hr : 0 < r) (hx : 0 < x) :
    0 < exponentialPDFReal r x 
goal: 0 < ProbabilityTheory.exponentialPDFReal r x

full_name: ProbabilityTheory.exponentialPDFReal_nonneg
code: /-- The exponential pdf is nonnegative-/
lemma exponentialPDFReal_nonneg {r : ℝ} (hr : 0 < r) (x : ℝ) :
    0 ≤ exponentialPDFReal r x 
goal: 0 ≤ ProbabilityTheory.exponentialPDFReal r x

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

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

full_name: ProbabilityTheory.isProbabilityMeasureExponential
code: lemma isProbabilityMeasureExponential {r : ℝ} (hr : 0 < r) :
    IsProbabilityMeasure (expMeasure r) 
goal: MeasureTheory.IsProbabilityMeasure (ProbabilityTheory.expMeasure r)

full_name: ProbabilityTheory.exponentialCDFReal
code: /-- CDF of the exponential distribution -/
noncomputable
def exponentialCDFReal (r : ℝ) : StieltjesFunction 
goal: StieltjesFunction

full_name: ProbabilityTheory.exponentialCDFReal_eq_integral
code: lemma exponentialCDFReal_eq_integral {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = ∫ x in Iic x, exponentialPDFReal r x 
goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = ∫ (x : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDFReal r x

full_name: ProbabilityTheory.exponentialCDFReal_eq_lintegral
code: lemma exponentialCDFReal_eq_lintegral {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = ENNReal.toReal (∫⁻ x in Iic x, exponentialPDF r x) 
goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = (∫⁻ (x : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDF r x).toReal

full_name: ProbabilityTheory.hasDerivAt_neg_exp_mul_exp
code: lemma hasDerivAt_neg_exp_mul_exp {r x : ℝ} :
    HasDerivAt (fun a ↦ -exp (-(r * a))) (r * exp (-(r * x))) x 
goal: HasDerivAt (fun a => -rexp (-(r * a))) (r * rexp (-(r * x))) x

full_name: ProbabilityTheory.exp_neg_integrableOn_Ioc
code: /-- A negative exponential function is integrable on intervals in `R≥0` -/
lemma exp_neg_integrableOn_Ioc {b x : ℝ} (hb : 0 < b) :
    IntegrableOn (fun x ↦ rexp (-(b * x))) (Ioc 0 x) 
goal: MeasureTheory.IntegrableOn (fun x => rexp (-(b * x))) (Set.Ioc 0 x) MeasureTheory.volume

full_name: ProbabilityTheory.lintegral_exponentialPDF_eq_antiDeriv
code: lemma lintegral_exponentialPDF_eq_antiDeriv {r : ℝ} (hr : 0 < r) (x : ℝ) :
    ∫⁻ y in Iic x, exponentialPDF r y
    = ENNReal.ofReal (if 0 ≤ x then 1 - exp (-(r * x)) else 0) 
goal: ∫⁻ (y : ℝ) in Set.Iic x, ProbabilityTheory.exponentialPDF r y =
  ENNReal.ofReal (if 0 ≤ x then 1 - rexp (-(r * x)) else 0)

full_name: ProbabilityTheory.exponentialCDFReal_eq
code: /-- The CDF of the exponential distribution equals ``1 - exp (-(r * x))``-/
lemma exponentialCDFReal_eq {r : ℝ} (hr : 0 < r) (x : ℝ) :
    exponentialCDFReal r x = if 0 ≤ x then 1 - exp (-(r * x)) else 0 
goal: ↑(ProbabilityTheory.exponentialCDFReal r) x = if 0 ≤ x then 1 - rexp (-(r * x)) else 0