Module mathcomp.analysis.probability

From mathcomp Require Import all_ssreflect.
From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality.
From HB Require Import structures.
Require Import exp numfun lebesgue_measure lebesgue_integral.
Require Import reals ereal signed topology normedtype sequences esum measure.
Require Import exp numfun lebesgue_measure lebesgue_integral.

# Probability This file provides basic notions of probability theory. See measure.v for the type probability T R (a measure that sums to 1). ``` {RV P >-> R} == real random variable: a measurable function from the measurableType of the probability P to R distribution X == measure image of P by X : {RV P -> R}, declared as an instance of probability measure 'E_P[X] == expectation of the real measurable function X covariance X Y == covariance between real random variable X and Y 'V_P[X] == variance of the real random variable X mmt_gen_fun X == moment generating function of the random variable X {dmfun T >-> R} == type of discrete real-valued measurable functions {dRV P >-> R} == real-valued discrete random variable dRV_dom X == domain of the discrete random variable X dRV_enum X == bijection between the domain and the range of X pmf X r := fine (P (X @^-1` [set r])) enum_prob X k == probability of the kth value in the range of X ```

Reserved Notation "'{' 'RV' P >-> R '}'"
  (at level 0, format "'{' 'RV'  P  '>->'  R '}'").
Reserved Notation "''E_' P [ X ]" (format "''E_' P [ X ]", at level 5).
Reserved Notation "''V_' P [ X ]" (format "''V_' P [ X ]", at level 5).
Reserved Notation "{ 'dmfun' aT >-> T }"
  (at level 0, format "{ 'dmfun'  aT  >->  T }").
Reserved Notation "'{' 'dRV' P >-> R '}'"
  (at level 0, format "'{' 'dRV'  P  '>->'  R '}'").

Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldTopology.Exports.

Local Open Scope classical_set_scope.
Local Open Scope ring_scope.

Definition random_variable ( d : _) ( T : measurableType d) ( R : realType)
  ( P : probability T R) := {mfun T >-> R}.

Notation "{ 'RV' P >-> R }" := (@random_variable _ _ R P) : form_scope.

Lemma notin_range_measure d ( T : measurableType d) ( R : realType)
    ( P : {measure set T -> \bar R}) ( X : T -> R) r :
  r \notin range X -> P (X @^-1` [set r]) = 0%E.
Proof.
by rewrite notin_set => hr; rewrite preimage10. Qed.

Lemma probability_range d ( T : measurableType d) ( R : realType)
  ( P : probability T R) ( X : {RV P >-> R}) : P (X @^-1` range X) = 1%E.
Proof.
by rewrite preimage_range probability_setT. Qed.

Definition distribution ( d : _) ( T : measurableType d) ( R : realType)
    ( P : probability T R) ( X : {mfun T >-> R}) :=
  pushforward P (@measurable_funP _ _ _ X).

Section distribution_is_probability .
Context d ( T : measurableType d) ( R : realType) ( P : probability T R)
        ( X : {mfun T >-> R}).

Let distribution0 : distribution P X set0 = 0%E.
Proof.
exact: measure0. Qed.

Let distribution_ge0 A : (0 <= distribution P X A)%E.
Proof.
exact: measure_ge0. Qed.

Let distribution_sigma_additive : semi_sigma_additive (distribution P X).
Proof.
exact: measure_semi_sigma_additive. Qed.

HB.instance Definition _ := isMeasure.Build _ _ R (distribution P X)
  distribution0 distribution_ge0 distribution_sigma_additive.

Let distribution_is_probability : distribution P X [set: _] = 1%:E.
Proof.
by rewrite /distribution /= /pushforward /= preimage_setT probability_setT.
Qed.

HB.instance Definition _ := Measure_isProbability.Build _ _ R
  (distribution P X) distribution_is_probability.

End distribution_is_probability.

Section transfer_probability .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Lemma probability_distribution ( X : {RV P >-> R}) r :
  P [set x | X x = r] = distribution P X [set r].
Proof.
by []. Qed.

Lemma integral_distribution ( X : {RV P >-> R}) ( f : R -> \bar R) :
    measurable_fun [set: R] f -> (forall y, 0 <= f y) ->
  \int[distribution P X]_y f y = \int[P]_x (f \o X) x.
Proof.
by move=> mf f0; rewrite integral_pushforward. Qed.

End transfer_probability.

HB.lock Definition expectation {d} {T : measurableType d} {R : realType}
  ( P : probability T R) ( X : T -> R) := (\int[P]_w (X w)%:E)%E.
Canonical expectation_unlockable := Unlockable expectation.unlock.
Arguments expectation {d T R} P _%R.
Notation "''E_' P [ X ]" := (@expectation _ _ _ P X) : ereal_scope.

Section expectation_lemmas .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Lemma expectation_fin_num ( X : {RV P >-> R}) : P.-integrable setT (EFin \o X) ->
  'E_P[X] \is a fin_num.
Proof.
by move=> ?; rewrite unlock integral_fune_fin_num. Qed.

Lemma expectation_cst r : 'E_P[cst r] = r%:E.
Proof.
by rewrite unlock/= integral_cst//= probability_setT mule1. Qed.

Lemma expectation_indic ( A : set T) ( mA : measurable A) : 'E_P[\1_A] = P A.
Proof.
by rewrite unlock integral_indic// setIT. Qed.

Lemma integrable_expectation ( X : {RV P >-> R})
  ( iX : P.-integrable [set: T] (EFin \o X)) : `| 'E_P[X] | < +oo.
Proof.
move: iX => /integrableP[? Xoo]; rewrite (le_lt_trans _ Xoo)// unlock.
exact: le_trans (le_abse_integral _ _ _).
Qed.

Lemma expectationM ( X : {RV P >-> R}) ( iX : P.-integrable [set: T] (EFin \o X))
  ( k : R) : 'E_P[k \o* X] = k%:E * 'E_P [X].
Proof.
rewrite unlock; under eq_integral do rewrite EFinM.
by rewrite -integralZl//; under eq_integral do rewrite muleC.
Qed.

Lemma expectation_ge0 ( X : {RV P >-> R}) :
  (forall x, 0 <= X x)%R -> 0 <= 'E_P[X].
Proof.
by move=> ?; rewrite unlock integral_ge0// => x _; rewrite lee_fin.
Qed.

Lemma expectation_le ( X Y : T -> R) :
    measurable_fun [set: T] X -> measurable_fun [set: T] Y ->
    (forall x, 0 <= X x)%R -> (forall x, 0 <= Y x)%R ->
  {ae P, (forall x, X x <= Y x)%R} -> 'E_P[X] <= 'E_P[Y].
Proof.
move=> mX mY X0 Y0 XY; rewrite unlock ae_ge0_le_integral => //.
- by move=> t _; apply: X0.
- exact/EFin_measurable_fun.
- by move=> t _; apply: Y0.
- exact/EFin_measurable_fun.
- move: XY => [N [mN PN XYN]]; exists N; split => // t /= h.
  by apply: XYN => /=; apply: contra_not h; rewrite lee_fin.
Qed.

Lemma expectationD ( X Y : {RV P >-> R}) :
    P.-integrable [set: T] (EFin \o X) -> P.-integrable [set: T] (EFin \o Y) ->
  'E_P[X \+ Y] = 'E_P[X] + 'E_P[Y].
Proof.
by move=> ? ?; rewrite unlock integralD_EFin. Qed.

Lemma expectationB ( X Y : {RV P >-> R}) :
    P.-integrable [set: T] (EFin \o X) -> P.-integrable [set: T] (EFin \o Y) ->
  'E_P[X \- Y] = 'E_P[X] - 'E_P[Y].
Proof.
by move=> ? ?; rewrite unlock integralB_EFin. Qed.

Lemma expectation_sum ( X : seq {RV P >-> R}) :
    (forall Xi, Xi \in X -> P.-integrable [set: T] (EFin \o Xi)) ->
  'E_P[\sum_( Xi <- X) Xi] = \sum_( Xi <- X) 'E_P[Xi].
Proof.
elim: X => [|X0 X IHX] intX; first by rewrite !big_nil expectation_cst.
have intX0 : P.-integrable [set: T] (EFin \o X0).
  by apply: intX; rewrite in_cons eqxx.
have {}intX Xi : Xi \in X -> P.-integrable [set: T] (EFin \o Xi).
  by move=> XiX; apply: intX; rewrite in_cons XiX orbT.
rewrite !big_cons expectationD ?IHX// (_ : _ \o _ = fun x =>
    \sum_( f <- map (fun x : {RV P >-> R} => EFin \o x) X) f x).
  by apply: integrable_sum => // _ /mapP[h hX ->]; exact: intX.
by apply/funext => t/=; rewrite big_map sumEFin mfun_sum.
Qed.

End expectation_lemmas.

HB.lock Definition covariance {d} {T : measurableType d} {R : realType}
    ( P : probability T R) ( X Y : T -> R) :=
  'E_P[(X \- cst (fine 'E_P[X])) * (Y \- cst (fine 'E_P[Y]))]%E.
Canonical covariance_unlockable := Unlockable covariance.unlock.
Arguments covariance {d T R} P _%R _%R.

Section covariance_lemmas .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Lemma covarianceE ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P X Y = 'E_P[X * Y] - 'E_P[X] * 'E_P[Y].
Proof.
move=> X1 Y1 XY1.
have ? : 'E_P[X] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
have ? : 'E_P[Y] \is a fin_num by rewrite fin_num_abs// integrable_expectation.
rewrite unlock [X in 'E_P[X]](_ : _ = (X \* Y \- fine 'E_P[X] \o* Y
    \- fine 'E_P[Y] \o* X \+ fine ('E_P[X] * 'E_P[Y]) \o* cst 1)%R); last first.
  apply/funeqP => x /=; rewrite mulrDr !mulrDl/= mul1r fineM// mulrNN addrA.
  by rewrite mulrN mulNr [Z in (X x * Y x - Z)%R]mulrC.
have ? : P.-integrable [set: T] (EFin \o (X \* Y \- fine 'E_P[X] \o* Y)%R).
  by rewrite compreBr ?integrableB// compre_scale ?integrableZl.
rewrite expectationD/=; last 2 first.
  - by rewrite compreBr// integrableB// compre_scale ?integrableZl.
  - by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
rewrite 2?expectationB//= ?compre_scale// ?integrableZl//.
rewrite 3?expectationM//= ?finite_measure_integrable_cst//.
by rewrite expectation_cst mule1 fineM// EFinM !fineK// muleC subeK ?fin_numM.
Qed.

Lemma covarianceC ( X Y : T -> R) : covariance P X Y = covariance P Y X.
Proof.
by rewrite unlock; congr expectation; apply/funeqP => x /=; rewrite mulrC.
Qed.

Lemma covariance_fin_num ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P X Y \is a fin_num.
Proof.
by move=> X1 Y1 XY1; rewrite covarianceE// fin_numB fin_numM expectation_fin_num.
Qed.

Lemma covariance_cst_l c ( X : {RV P >-> R}) : covariance P (cst c) X = 0.
Proof.
rewrite unlock expectation_cst/=.
rewrite [X in 'E_P[X]](_ : _ = cst 0%R) ?expectation_cst//.
by apply/funeqP => x; rewrite /GRing.mul/= subrr mul0r.
Qed.

Lemma covariance_cst_r ( X : {RV P >-> R}) c : covariance P X (cst c) = 0.
Proof.
by rewrite covarianceC covariance_cst_l. Qed.

Lemma covarianceZl a ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P (a \o* X)%R Y = a%:E * covariance P X Y.
Proof.
move=> X1 Y1 XY1.
have aXY : (a \o* X * Y = a \o* (X * Y))%R.
  by apply/funeqP => x; rewrite mulrAC.
rewrite [LHS]covarianceE => [||//|] /=; last 2 first.
- by rewrite compre_scale ?integrableZl.
- by rewrite aXY compre_scale ?integrableZl.
rewrite covarianceE// aXY !expectationM//.
by rewrite -muleA -muleBr// fin_num_adde_defr// expectation_fin_num.
Qed.

Lemma covarianceZr a ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P X (a \o* Y)%R = a%:E * covariance P X Y.
Proof.
move=> X1 Y1 XY1.
by rewrite [in RHS]covarianceC covarianceC covarianceZl; last rewrite mulrC.
Qed.

Lemma covarianceNl ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P (\- X)%R Y = - covariance P X Y.
Proof.
move=> X1 Y1 XY1.
have -> : (\- X = -1 \o* X)%R by apply/funeqP => x /=; rewrite mulrN mulr1.
by rewrite covarianceZl// EFinN mulNe mul1e.
Qed.

Lemma covarianceNr ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P X (\- Y)%R = - covariance P X Y.
Proof.
by move=> X1 Y1 XY1; rewrite !(covarianceC X) covarianceNl 1?mulrC. Qed.

Lemma covarianceNN ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) ->
    P.-integrable setT (EFin \o Y) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P (\- X)%R (\- Y)%R = covariance P X Y.
Proof.
move=> X1 Y1 XY1.
have NY : P.-integrable setT (EFin \o (\- Y)%R) by rewrite compreN ?integrableN.
by rewrite covarianceNl ?covarianceNr ?oppeK//= mulrN compreN ?integrableN.
Qed.

Lemma covarianceDl ( X Y Z : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Z)%R) ->
    P.-integrable setT (EFin \o (Y * Z)%R) ->
  covariance P (X \+ Y)%R Z = covariance P X Z + covariance P Y Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XZ1 YZ1.
rewrite [LHS]covarianceE//= ?mulrDl ?compreDr// ?integrableD//.
rewrite 2?expectationD//=.
rewrite muleDl ?fin_num_adde_defr ?expectation_fin_num//.
rewrite oppeD ?fin_num_adde_defr ?fin_numM ?expectation_fin_num//.
by rewrite addeACA 2?covarianceE.
Qed.

Lemma covarianceDr ( X Y Z : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
    P.-integrable setT (EFin \o (X * Z)%R) ->
  covariance P X (Y \+ Z)%R = covariance P X Y + covariance P X Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XY1 XZ1.
by rewrite covarianceC covarianceDl ?(covarianceC X) 1?mulrC.
Qed.

Lemma covarianceBl ( X Y Z : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Z)%R) ->
    P.-integrable setT (EFin \o (Y * Z)%R) ->
  covariance P (X \- Y)%R Z = covariance P X Z - covariance P Y Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XZ1 YZ1.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R covarianceDl ?covarianceNl//=.
- by rewrite compreN// integrableN.
- by rewrite mulrNN.
- by rewrite mulNr compreN// integrableN.
Qed.

Lemma covarianceBr ( X Y Z : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o Z) -> P.-integrable setT (EFin \o (Z ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
    P.-integrable setT (EFin \o (X * Z)%R) ->
  covariance P X (Y \- Z)%R = covariance P X Y - covariance P X Z.
Proof.
move=> X1 X2 Y1 Y2 Z1 Z2 XY1 XZ1.
by rewrite !(covarianceC X) covarianceBl 1?(mulrC _ X).
Qed.

End covariance_lemmas.

Section variance .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Definition variance ( X : T -> R) := covariance P X X.
Local Notation "''V_' P [ X ]" := (variance X).

Lemma varianceE ( X : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[X] = 'E_P[X ^+ 2] - ('E_P[X]) ^+ 2.
Proof.
by move=> X1 X2; rewrite /variance covarianceE. Qed.

Lemma variance_fin_num ( X : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o X ^+ 2)%R ->
  'V_P[X] \is a fin_num.
Proof.
by move=> /[dup]; apply: covariance_fin_num. Qed.

Lemma variance_ge0 ( X : {RV P >-> R}) : (0 <= 'V_P[X])%E.
Proof.
by rewrite /variance unlock; apply: expectation_ge0 => x; apply: sqr_ge0.
Qed.

Lemma variance_cst r : 'V_P[cst r] = 0%E.
Proof.
rewrite /variance unlock expectation_cst/=.
rewrite [X in 'E_P[X]](_ : _ = cst 0%R) ?expectation_cst//.
by apply/funext => x; rewrite /GRing.exp/GRing.mul/= subrr mulr0.
Qed.

Lemma varianceZ a ( X : {RV P >-> R}) :
  P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(a \o* X)%R] = (a ^+ 2)%:E * 'V_P[X].
Proof.
move=> X1 X2; rewrite /variance covarianceZl//=.
- by rewrite covarianceZr// muleA.
- by rewrite compre_scale// integrableZl.
- rewrite [X in EFin \o X](_ : _ = (a \o* X ^+ 2)%R); last first.
    by apply/funeqP => x; rewrite mulrA.
  by rewrite compre_scale// integrableZl.
Qed.

Lemma varianceN ( X : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(\- X)%R] = 'V_P[X].
Proof.
by move=> X1 X2; rewrite /variance covarianceNN. Qed.

Lemma varianceD ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  'V_P[X \+ Y]%R = 'V_P[X] + 'V_P[Y] + 2%:E * covariance P X Y.
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -['V_P[_]]/(covariance P (X \+ Y)%R (X \+ Y)%R).
have XY : P.-integrable [set: T] (EFin \o (X \+ Y)%R).
  by rewrite compreDr// integrableD.
rewrite covarianceDl//=; last 3 first.
- rewrite -expr2 sqrrD compreDr ?integrableD// compreDr// integrableD//.
  rewrite -mulr_natr -[(_ * 2)%R]/(2 \o* (X * Y))%R compre_scale//.
  exact: integrableZl.
- by rewrite mulrDr compreDr ?integrableD.
- by rewrite mulrDr mulrC compreDr ?integrableD.
rewrite covarianceDr// covarianceDr ?(mulrC Y X)//.
rewrite (covarianceC P Y X) [LHS]addeA [LHS](ACl (1*4*(2*3)))/=.
by rewrite -[2%R]/(1 + 1)%R EFinD muleDl ?mul1e// covariance_fin_num.
Qed.

Lemma varianceB ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  'V_P[(X \- Y)%R] = 'V_P[X] + 'V_P[Y] - 2%:E * covariance P X Y.
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -[(X \- Y)%R]/(X \+ (\- Y))%R.
rewrite varianceD/= ?varianceN ?covarianceNr ?muleN//.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN.
- by rewrite mulrN compreN ?integrableN.
Qed.

Lemma varianceD_cst_l c ( X : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(cst c \+ X)%R] = 'V_P[X].
Proof.
move=> X1 X2.
rewrite varianceD//=; last 3 first.
- exact: finite_measure_integrable_cst.
- by rewrite compre_scale// integrableZl// finite_measure_integrable_cst.
- by rewrite mulrC compre_scale ?integrableZl.
by rewrite variance_cst add0e covariance_cst_l mule0 adde0.
Qed.

Lemma varianceD_cst_r ( X : {RV P >-> R}) c :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(X \+ cst c)%R] = 'V_P[X].
Proof.
move=> X1 X2.
have -> : (X \+ cst c = cst c \+ X)%R by apply/funeqP => x /=; rewrite addrC.
exact: varianceD_cst_l.
Qed.

Lemma varianceB_cst_l c ( X : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(cst c \- X)%R] = 'V_P[X].
Proof.
move=> X1 X2.
rewrite -[(cst c \- X)%R]/(cst c \+ (\- X))%R varianceD_cst_l/=; last 2 first.
- by rewrite compreN ?integrableN.
- by rewrite mulrNN; apply: X2.
by rewrite varianceN.
Qed.

Lemma varianceB_cst_r ( X : {RV P >-> R}) c :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
  'V_P[(X \- cst c)%R] = 'V_P[X].
Proof.
by move=> X1 X2; rewrite -[(X \- cst c)%R]/(X \+ (cst (- c)))%R varianceD_cst_r.
Qed.

Lemma covariance_le ( X Y : {RV P >-> R}) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    P.-integrable setT (EFin \o Y) -> P.-integrable setT (EFin \o (Y ^+ 2)%R) ->
    P.-integrable setT (EFin \o (X * Y)%R) ->
  covariance P X Y <= sqrte 'V_P[X] * sqrte 'V_P[Y].
Proof.
move=> X1 X2 Y1 Y2 XY1.
rewrite -sqrteM ?variance_ge0//.
rewrite lee_sqrE ?sqrte_ge0// sqr_sqrte ?mule_ge0 ?variance_ge0//.
rewrite -(fineK (variance_fin_num X1 X2)) -(fineK (variance_fin_num Y1 Y2)).
rewrite -(fineK (covariance_fin_num X1 Y1 XY1)).
rewrite -EFin_expe -EFinM lee_fin -(@ler_pmul2l _ 4%:R) ?ltr0n// [leRHS]mulrA.
rewrite [in leLHS](natrM _ 2 2) mulrACA -expr2 -subr_le0.
apply: deg_le2_ge0 => r; rewrite -lee_fin !EFinD.
rewrite EFinM fineK ?variance_fin_num// muleC -varianceZ//.
rewrite -mulrA EFinM mulrC EFinM ?fineK ?covariance_fin_num// -covarianceZl//.
rewrite addeAC -varianceD ?variance_ge0//=.
- by rewrite compre_scale ?integrableZl.
- rewrite [X in EFin \o X](_ : _ = r ^+2 \o* X ^+ 2)%R 1?mulrACA//.
  by rewrite compre_scale ?integrableZl.
- by rewrite -mulrAC compre_scale// integrableZl.
Qed.

End variance.
Notation "'V_ P [ X ]" := (variance P X).

Section markov_chebyshev_cantelli .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Lemma markov ( X : {RV P >-> R}) ( f : R -> R) ( eps : R) :
    (0 < eps)%R ->
    measurable_fun [set: R] f -> (forall r, 0 <= f r)%R ->
    {in Num.nneg &, {homo f : x y / x <= y}}%R ->
  (f eps)%:E * P [set x | eps%:E <= `| (X x)%:E | ] <=
    'E_P[f \o (fun x => `| x |%R) \o X].
Proof.
move=> e0 mf f0 f_nd; rewrite -(setTI [set _ | _]).
apply: (le_trans (@le_integral_comp_abse _ _ _ P _ measurableT (EFin \o X)
  eps (er_map f) _ _ _ _ e0)) => //=.
- exact: measurable_er_map.
- by case => //= r _; exact: f0.
- move=> [x| |] [y| |]; rewrite !inE/= !in_itv/= ?andbT ?lee_fin ?leey//.
  by move=> ? ? ?; rewrite f_nd.
- exact/EFin_measurable_fun.
- by rewrite unlock.
Qed.

Definition mmt_gen_fun ( X : {RV P >-> R}) ( t : R) := 'E_P[expR \o t \o* X].

Lemma chernoff ( X : {RV P >-> R}) ( r a : R) : (0 < r)%R ->
  P [set x | X x >= a]%R <= mmt_gen_fun X r * (expR (- (r * a)))%:E.
Proof.
move=> t0.
rewrite /mmt_gen_fun; have -> : expR \o r \o* X =
    (normr \o normr) \o [the {mfun T >-> R} of expR \o r \o* X].
  by apply: funext => t /=; rewrite normr_id ger0_norm ?expR_ge0.
rewrite expRN lee_pdivl_mulr ?expR_gt0//.
rewrite (le_trans _ (markov _ (expR_gt0 (r * a)) _ _ _))//; last first.
  exact: (monoW_in (@ger0_le_norm _)).
rewrite ger0_norm ?expR_ge0// muleC lee_pmul2l// ?lte_fin ?expR_gt0//.
rewrite [X in _ <= P X](_ : _ = [set x | a <= X x]%R)//; apply: eq_set => t/=.
by rewrite ger0_norm ?expR_ge0// lee_fin ler_expR mulrC ler_pmul2r.
Qed.

Lemma chebyshev ( X : {RV P >-> R}) ( eps : R) : (0 < eps)%R ->
  P [set x | (eps <= `| X x - fine ('E_P[X])|)%R ] <= (eps ^- 2)%:E * 'V_P[X].
Proof.
move => heps; have [->|hv] := eqVneq 'V_P[X] +oo.
  by rewrite mulr_infty gtr0_sg ?mul1e// ?leey// invr_gt0// exprn_gt0.
have h ( Y : {RV P >-> R}) :
    P [set x | (eps <= `|Y x|)%R] <= (eps ^- 2)%:E * 'E_P[Y ^+ 2].
  rewrite -lee_pdivr_mull; last by rewrite invr_gt0// exprn_gt0.
  rewrite exprnN expfV exprz_inv opprK -exprnP.
  apply: (@le_trans _ _ ('E_P[(@GRing.exp R ^~ 2%N \o normr) \o Y])).
    apply: (@markov Y (@GRing.exp R ^~ 2%N)) => //.
    - by move=> r; apply: sqr_ge0.
    - move=> x y; rewrite !nnegrE => x0 y0.
      by rewrite ler_sqr.
  apply: expectation_le => //.
    - by apply: measurableT_comp => //; exact: measurableT_comp.
  - by move=> x /=; apply: sqr_ge0.
  - by move=> x /=; apply: sqr_ge0.
  - by apply/aeW => t /=; rewrite real_normK// num_real.
have := h [the {mfun T >-> R} of (X \- cst (fine ('E_P[X])))%R].
by move=> /le_trans; apply; rewrite /variance [in leRHS]unlock.
Qed.

Lemma cantelli ( X : {RV P >-> R}) ( lambda : R) :
    P.-integrable setT (EFin \o X) -> P.-integrable setT (EFin \o (X ^+ 2)%R) ->
    (0 < lambda)%R ->
  P [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
  <= (fine 'V_P[X] / (fine 'V_P[X] + lambda^2))%:E.
Proof.
move=> X1 X2 lambda_gt0.
have finEK : (fine 'E_P[X])%:E = 'E_P[X].
  by rewrite fineK ?unlock ?integral_fune_fin_num.
have finVK : (fine 'V_P[X])%:E = 'V_P[X] by rewrite fineK ?variance_fin_num.
pose Y := (X \- cst (fine 'E_P[X]))%R.
have Y1 : P.-integrable [set: T] (EFin \o Y).
  rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
  exact: finite_measure_integrable_cst.
have Y2 : P.-integrable [set: T] (EFin \o (Y ^+ 2)%R).
  rewrite sqrrD/= compreDr => [|//].
  apply: integrableD => [//||]; last first.
    rewrite -[(_ ^+ 2)%R]/(cst ((- fine 'E_P[X]) ^+ 2)%R).
    exact: finite_measure_integrable_cst.
  rewrite compreDr => [|//]; apply: integrableD X2 _ => [//|].
  rewrite [X in EFin \o X](_ : _ = (- fine 'E_P[X] * 2) \o* X)%R; last first.
    by apply/funeqP => x /=; rewrite -mulr_natl mulrC mulrA.
  by rewrite compre_scale => [|//]; apply: integrableZl X1.
have EY : 'E_P[Y] = 0.
  rewrite expectationB/= ?finite_measure_integrable_cst//.
  rewrite expectation_cst finEK subee//.
  by rewrite unlock; apply: integral_fune_fin_num X1.
have VY : 'V_P[Y] = 'V_P[X] by rewrite varianceB_cst_r.
have le ( u : R) : (0 <= u)%R ->
    P [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
    <= ((fine 'V_P[X] + u^2) / (lambda + u)^2)%:E.
  move=> uge0; rewrite EFinM.
  have YU1 : P.-integrable [set: T] (EFin \o (Y \+ cst u)%R).
    rewrite compreDr => [|//]; apply: integrableD Y1 _ => [//|].
    exact: finite_measure_integrable_cst.
  have YU2 : P.-integrable [set: T] (EFin \o ((Y \+ cst u) ^+ 2)%R).
    rewrite sqrrD/= compreDr => [|//].
    apply: integrableD => [//||]; last first.
      rewrite -[(_ ^+ 2)%R]/(cst (u ^+ 2))%R.
      exact: finite_measure_integrable_cst.
    rewrite compreDr => [|//]; apply: integrableD Y2 _ => [//|].
    rewrite [X in EFin \o X](_ : _ = (2 * u) \o* Y)%R; last first.
      by apply/funeqP => x /=; rewrite -mulr_natl mulrCA.
    by rewrite compre_scale => [|//]; apply: integrableZl Y1.
  have -> : (fine 'V_P[X] + u^2)%:E = 'E_P[(Y \+ cst u)^+2]%R.
    rewrite -VY -[RHS](@subeK _ _ (('E_P[(Y \+ cst u)%R])^+2)); last first.
      by rewrite fin_numX ?unlock ?integral_fune_fin_num.
    rewrite -varianceE/= -/Y -?expe2//.
    rewrite expectationD/= ?EY ?add0e ?expectation_cst -?EFinM; last 2 first.
    - rewrite compreBr => [|//]; apply: integrableB X1 _ => [//|].
      exact: finite_measure_integrable_cst.
    - exact: finite_measure_integrable_cst.
    by rewrite (varianceD_cst_r _ Y1 Y2) EFinD fineK ?(variance_fin_num Y1 Y2).
  have le : [set x | lambda%:E <= (X x)%:E - 'E_P[X]]
      `<=` [set x | ((lambda + u)^2)%:E <= ((Y x + u)^+2)%:E].
    move=> x /= le; rewrite lee_fin; apply: ler_expn2r.
    - exact: addr_ge0 (ltW lambda_gt0) _.
    - apply/(addr_ge0 _ uge0)/(le_trans (ltW lambda_gt0) _).
      by rewrite -lee_fin EFinB finEK.
    - by rewrite ler_add2r -lee_fin EFinB finEK.
  apply: (le_trans (le_measure _ _ _ le)).
  - rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
    by apply: emeasurable_funB => //; exact: measurable_int X1.
  - rewrite -[[set _ | _]]setTI inE; apply: emeasurable_fun_c_infty => [//|].
    rewrite EFin_measurable_fun [X in measurable_fun _ X](_ : _ =
      (fun x => x ^+ 2) \o (fun x => Y x + u))%R//.
    apply/measurableT_comp => //; apply/measurable_funD => //.
    by rewrite -EFin_measurable_fun; apply: measurable_int Y1.
  set eps := ((lambda + u) ^ 2)%R.
  have peps : (0 < eps)%R by rewrite exprz_gt0 ?ltr_paddr.
  rewrite (lee_pdivl_mulr _ _ peps) muleC.
  under eq_set => x.
    rewrite -[leRHS]gee0_abs ?lee_fin ?sqr_ge0 -?lee_fin => [|//].
    rewrite -[(_ ^+ 2)%R]/(((Y \+ cst u) ^+ 2) x)%R; over.
  rewrite -[X in X%:E * _]gtr0_norm => [|//].
  apply: (le_trans (markov _ peps _ _ _)) => //=.
    by move=> x y /[!nnegrE] /ger0_norm-> /ger0_norm->.
  rewrite -/Y le_eqVlt; apply/orP; left; apply/eqP; congr expectation.
  by apply/funeqP => x /=; rewrite -expr2 normr_id ger0_norm ?sqr_ge0.
pose u0 := (fine 'V_P[X] / lambda)%R.
have u0ge0 : (0 <= u0)%R.
  by apply: divr_ge0 (ltW lambda_gt0); rewrite -lee_fin finVK variance_ge0.
apply: le_trans (le _ u0ge0) _; rewrite lee_fin le_eqVlt; apply/orP; left.
rewrite GRing.eqr_div; [|apply: lt0r_neq0..]; last 2 first.
- by rewrite exprz_gt0 -1?[ltLHS]addr0 ?ltr_le_add.
- by rewrite ltr_paddl ?fine_ge0 ?variance_ge0 ?exprz_gt0.
apply/eqP; have -> : fine 'V_P[X] = (u0 * lambda)%R.
  by rewrite /u0 -mulrA mulVr ?mulr1 ?unitfE ?gt_eqF.
by rewrite -mulrDl -mulrDr (addrC u0) [in RHS](mulrAC u0) -exprnP expr2 !mulrA.
Qed.

End markov_chebyshev_cantelli.

HB.mixin Record MeasurableFun_isDiscrete d ( T : measurableType d) ( R : realType)
    ( X : T -> R) of @MeasurableFun d T R X := {
   countable_range : countable (range X)
}.

HB.structure Definition discreteMeasurableFun d ( T : measurableType d)
    ( R : realType) := {
   X of isMeasurableFun d T R X & MeasurableFun_isDiscrete d T R X
}.

Notation "{ 'dmfun' aT >-> T }" :=
  (@discreteMeasurableFun.type _ aT T) : form_scope.

Definition discrete_random_variable ( d : _) ( T : measurableType d)
  ( R : realType) ( P : probability T R) := {dmfun T >-> R}.

Notation "{ 'dRV' P >-> R }" :=
  (@discrete_random_variable _ _ R P) : form_scope.

Section dRV_definitions .
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Definition dRV_dom_enum ( X : {dRV P >-> R}) :
  { B : set nat & {splitbij B >-> range X}}.
Proof.
have /countable_bijP/cid[B] := @countable_range _ _ _ X.
move/card_esym/ppcard_eqP/unsquash => f.
exists B; exact: f.
Qed.

Definition dRV_dom ( X : {dRV P >-> R}) : set nat := projT1 (dRV_dom_enum X).

Definition dRV_enum ( X : {dRV P >-> R}) : {splitbij (dRV_dom X) >-> range X} :=
  projT2 (dRV_dom_enum X).

Definition enum_prob ( X : {dRV P >-> R}) :=
  (fun k => P (X @^-1` [set dRV_enum X k])) \_ (dRV_dom X).

End dRV_definitions.

Section distribution_dRV .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).
Variable X : {dRV P >-> R}.

Lemma distribution_dRV_enum ( n : nat) : n \in dRV_dom X ->
  distribution P X [set dRV_enum X n] = enum_prob X n.
Proof.
by move=> nX; rewrite /distribution/= /enum_prob/= patchE nX.
Qed.

Lemma distribution_dRV A : measurable A ->
  distribution P X A = \sum_( k <oo) enum_prob X k * \d_ (dRV_enum X k) A.
Proof.
move=> mA; rewrite /distribution /pushforward.
have mAX i : dRV_dom X i -> measurable (X @^-1` (A `&` [set dRV_enum X i])).
  move=> _; rewrite preimage_setI; apply: measurableI => //.
  exact/measurable_sfunP.
have tAX : trivIset (dRV_dom X) (fun k => X @^-1` (A `&` [set dRV_enum X k])).
  under eq_fun do rewrite preimage_setI; rewrite -/(trivIset _ _).
  apply: trivIset_setIl; apply/trivIsetP => i j iX jX /eqP ij.
  rewrite -preimage_setI (_ : _ `&` _ = set0)//.
  by apply/seteqP; split => //= x [] -> {x} /inj; rewrite inE inE => /(_ iX jX).
have := measure_bigcup P _ (fun k => X @^-1` (A `&` [set dRV_enum X k])) mAX tAX.
rewrite -preimage_bigcup => {mAX tAX}PXU.
rewrite -{1}(setIT A) -(setUv (\bigcup_( i in dRV_dom X) [set dRV_enum X i])).
rewrite setIUr preimage_setU measureU; last 3 first.
  - rewrite preimage_setI; apply: measurableI => //.
      exact: measurable_sfunP.
    by apply: measurable_sfunP; exact: bigcup_measurable.
  - apply: measurable_sfunP; apply: measurableI => //.
    by apply: measurableC; exact: bigcup_measurable.
  - rewrite 2!preimage_setI setIACA -!setIA -preimage_setI.
    by rewrite setICr preimage_set0 2!setI0.
rewrite [X in _ + X = _](_ : _ = 0) ?adde0; last first.
  rewrite (_ : _ @^-1` _ = set0) ?measure0//; apply/disjoints_subset => x AXx.
  rewrite setCK /bigcup /=; exists ((dRV_enum X)^-1 (X x))%function.
    exact: funS.
  by rewrite invK// inE.
rewrite setI_bigcupr; etransitivity; first exact: PXU.
rewrite eseries_mkcond; apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => nX; rewrite ?mul0e//.
rewrite diracE; have [kA|] := boolP (_ \in A).
  by rewrite mule1 setIidr// => _ /= ->; exact: set_mem.
rewrite notin_set => kA.
rewrite mule0 (disjoints_subset _ _).2 ?preimage_set0 ?measure0//.
by apply: subsetCr; rewrite sub1set inE.
Qed.

Lemma sum_enum_prob : \sum_( n <oo) (enum_prob X) n = 1.
Proof.
have := distribution_dRV measurableT.
rewrite probability_setT/= => /esym; apply: eq_trans.
by rewrite [RHS]eseries_mkcond; apply: eq_eseriesr => k _; rewrite diracT mule1.
Qed.

End distribution_dRV.

Section discrete_distribution .
Local Open Scope ereal_scope.
Context d ( T : measurableType d) ( R : realType) ( P : probability T R).

Lemma dRV_expectation ( X : {dRV P >-> R}) :
  P.-integrable [set: T] (EFin \o X) ->
  'E_P[X] = \sum_( n <oo) enum_prob X n * (dRV_enum X n)%:E.
Proof.
move=> ix; rewrite unlock.
rewrite -[in LHS](_ : \bigcup_k (if k \in dRV_dom X then
    X @^-1` [set dRV_enum X k] else set0) = setT); last first.
  apply/seteqP; split => // t _.
  exists ((dRV_enum X)^-1%function (X t)) => //.
  case: ifPn=> [_|].
    by rewrite invK// inE.
  by rewrite notin_set/=; apply; apply: funS.
have tA : trivIset (dRV_dom X) (fun k => [set dRV_enum X k]).
  by move=> i j iX jX [r [/= ->{r}]] /inj; rewrite !inE; exact.
have {tA}/trivIset_mkcond tXA :
    trivIset (dRV_dom X) (fun k => X @^-1` [set dRV_enum X k]).
  apply/trivIsetP => /= i j iX jX ij.
  move/trivIsetP : tA => /(_ i j iX jX) Aij.
  by rewrite -preimage_setI Aij ?preimage_set0.
rewrite integral_bigcup //; last 2 first.
  - by move=> k; case: ifPn.
  - apply: (integrableS measurableT) => //.
    by rewrite -bigcup_mkcond; exact: bigcup_measurable.
transitivity (\sum_( i <oo)
  \int[P]_( x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
    (dRV_enum X i)%:E).
  apply: eq_eseriesr => i _; case: ifPn => iX.
    by apply: eq_integral => t; rewrite in_setE/= => ->.
  by rewrite !integral_set0.
transitivity (\sum_( i <oo) (dRV_enum X i)%:E *
  \int[P]_( x in (if i \in dRV_dom X then X @^-1` [set dRV_enum X i] else set0))
    1).
  apply: eq_eseriesr => i _; rewrite -integralZl//; last 2 first.
    - by case: ifPn.
    - apply/integrableP; split => //.
      rewrite (eq_integral (cst 1%E)); last by move=> x _; rewrite abse1.
      rewrite integral_cst//; last by case: ifPn.
      rewrite mul1e (@le_lt_trans _ _ 1%E) ?ltey//.
      by case: ifPn => // _; exact: probability_le1.
  by apply: eq_integral => y _; rewrite mule1.
apply: eq_eseriesr => k _; case: ifPn => kX.
  rewrite /= integral_cst//= mul1e probability_distribution muleC.
  by rewrite distribution_dRV_enum.
by rewrite integral_set0 mule0 /enum_prob patchE (negbTE kX) mul0e.
Qed.

Definition pmf ( X : {RV P >-> R}) ( r : R) : R := fine (P (X @^-1` [set r])).

Lemma expectation_pmf ( X : {dRV P >-> R}) :
    P.-integrable [set: T] (EFin \o X) -> 'E_P[X] =
  \sum_( n <oo | n \in dRV_dom X) (pmf X (dRV_enum X n))%:E * (dRV_enum X n)%:E.
Proof.
move=> iX; rewrite dRV_expectation// [in RHS]eseries_mkcond.
apply: eq_eseriesr => k _.
rewrite /enum_prob patchE; case: ifPn => kX; last by rewrite mul0e.
by rewrite /pmf fineK// fin_num_measure.
Qed.

End discrete_distribution.
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