proposition: 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: 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.
imports: mathcomp(all_ssreflect) mathcomp(ssralg poly ssrnum ssrint interval finmap) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality) HB(structures) exp numfun lebesgue_measure lebesgue_integral reals ereal signed topology normedtype sequences esum measure exp numfun lebesgue_measure lebesgue_integral