← Mathematical handbook

Chatterjee xi of the two-parameter Marshall–Olkin family

theorem Verification.integral_marshallOlkinConditional_sq_response (α β u : ↑unitInterval) (hu : 0 < u) (hb : 0 < ↑β) : ∫ (v : ↑unitInterval), marshallOlkinConditional α β u v ^ 2 = (1 - ↑α) ^ 2 * ↑u ^ (↑α * (3 - 2 * ↑β) / ↑β) / 3 + (1 - ↑u ^ (↑α * (3 - 2 * ↑β) / ↑β)) / (3 - 2 * ↑β)

theorem Verification.marshallOlkin_chatterjeeXi_pos (α β : ↑unitInterval) (ha : 0 < ↑α) (hb : 0 < ↑β) : (ProbabilityTheory.Copula.marshallOlkin α β).chatterjeeXi = 2 * ↑α ^ 2 * ↑β / (3 * ↑α + ↑β - 2 * ↑α * ↑β)

theorem Verification.marshallOlkin_chatterjeeXi (α β : ↑unitInterval) : (ProbabilityTheory.Copula.marshallOlkin α β).chatterjeeXi = 2 * ↑α ^ 2 * ↑β / (3 * ↑α + ↑β - 2 * ↑α * ↑β)