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Tail expectation formula for L1 random variable #1865

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2 changes: 1 addition & 1 deletion CHANGELOG_UNRELEASED.md
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Expand Up @@ -43,7 +43,7 @@

- in `random_variable.v`
+ lemmas `lebesgue_integral_pmf`, `cdf_measurable`, `ccdf_measurable`,
`le0_expectation_cdf`
`le0_expectation_cdf`, `expectation_cdf_ccdf`

- in `lebesgue_integral_nonneg.v`:
+ lemma `integral_setU`
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31 changes: 31 additions & 0 deletions theories/probability_theory/random_variable.v
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Expand Up @@ -687,6 +687,37 @@ rewrite setDitv1r; congr -%E; apply: eq_integral => r _.
by rewrite (@comp_preimage _ _ _ _ _ -%R)/= opp_preimage_itvbndy/= opprK.
Qed.

Lemma expectation_cdf_ccdf (X : {RV P >-> R}) : Lfun P 1 X ->
'E_P[X] = \int[mu]_(r in `[0%R, +oo[) ccdf X r
- \int[mu]_(r in `]-oo, 0%R[) cdf X r.
Proof.
move=> L1X.
pose Xp := (cst 0%R) \max X.
pose Xm := (cst 0%R) \min X.
have Xp_leX x : setT x -> `|(Xp x)%:E| <= `|(X x)%:E|.
by rewrite lee_tofin// ger0_norm ?le_max ?lexx// ge_max normr_ge0 ler_norm.
have Xm_leX x : setT x -> `|(Xm x)%:E| <= `|(X x)%:E|.
rewrite /Xm lee_tofin//=.
by have [//=| Xle0 |] := (@real_le0P _ (X x));
[rewrite ler0_norm | rewrite normr0; exact: ltW].
have XpP1 : Xp \in Lfun P 1.
by rewrite Lfun1_integrable; apply: (le_integrable _ _ Xp_leX);
[| apply/measurable_EFinP | rewrite -Lfun1_integrable].
have XmP1: Xm \in Lfun P 1.
by rewrite Lfun1_integrable; apply: (le_integrable _ _ Xm_leX);
[| apply/measurable_EFinP | rewrite -Lfun1_integrable].
rewrite (_ : MeasurableFun.sort X = (Xp \+ Xm)%R) ?expectationD//;
last by apply: funext => x; rewrite /= addr_max_min add0r.
rewrite (_ : 'E_P[Xp] = \int[mu]_(r in `[0%R, +oo[) ccdf X r); last first.
rewrite ge0_expectation_ccdf/= => [|x]; last by case: ler0P; last exact: ltW.
apply: eq_integral => r; rewrite inE/= in_itv/= andbT => r_ge0; congr (P _).
by apply: eq_set => x/=; rewrite !in_itv/= !andbT lt_max ltNge r_ge0.
rewrite (_ : 'E_P[Xm] = - \int[mu]_(r in `]-oo, 0%R[) cdf X r)//.
rewrite le0_expectation_cdf/= => [|x]; last by rewrite ge_min ?lexx.
congr -%E; apply: eq_integral => r; rewrite inE/= in_itv/= => r_le0.
by congr (P _); apply: eq_set => x/=; rewrite !in_itv/= ge_min leNgt r_le0.
Qed.

End tail_expectation_formula.

HB.lock Definition covariance {d} {T : measurableType d} {R : realType}
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