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Both directional coefficients are affine under a tagged predictor join

theorem Verification.conditionalJoin_xi (C D : ProbabilityTheory.Copula 2) (a : ↑unitInterval) (ha0 : 0 < a) (ha1 : a < 1) : (conditionalJoin C D a ha0 ha1).chatterjeeXi = ↑a * C.chatterjeeXi + (1 - ↑a) * D.chatterjeeXi

theorem Verification.conditionalMean_conditionalJoin (C D : ProbabilityTheory.Copula 2) (a : ↑unitInterval) (ha0 : 0 < a) (ha1 : a < 1) : conditionalMean (conditionalJoin C D a ha0 ha1) =ᵐ[MeasureTheory.volume] unitJoin a (conditionalMean C) (conditionalMean D)

theorem Verification.conditionalJoin_correlationRatio (C D : ProbabilityTheory.Copula 2) (a : ↑unitInterval) (ha0 : 0 < a) (ha1 : a < 1) : correlationRatio (conditionalJoin C D a ha0 ha1) = ↑a * correlationRatio C + (1 - ↑a) * correlationRatio D