proposition: 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: Proof. move=> X1 X2 Y1 Y2 Z1 Z2 XY1 XZ1. by rewrite covarianceC covarianceDl ?(covarianceC X) 1?mulrC. 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