Part 1. Sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ Finally, the lower bound P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) can be obtained by taking the intersection over all z ∈ Ξ , 40 P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) ≥ sup z ∈ Ξ $^{{}$P ( y$_{0}$ | 0 , z ) L wst 10 ( y$_{1}$, z ) + L wst 01 ( y$_{1}$, z ) P ( y$_{1}$ | 1 , z ) $^{}}$. P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) = P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | z ) = P ( Y ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) (1 − p ( z )) + P ( Y$_{0}$ ≤ y$_{0}$, Y ≤ y$_{1}$ | 1 , z ) p ( z ) ≥ P ( y$_{0}$ | 0 , z ) L wst 10 ( y$_{1}$, z ) + L wst 01 ( y$_{1}$, z ) P ( y$_{1}$ | 1 , z ) P ( Y ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) ≥ ( 1 1 − p ( z )$^{)}$$^{2}$ ( Y$_{0}$ ≤ y$_{0}$ | 0 , z ) P ( Y$_{1}$ ≤ y$_{1}$ | 0 , z ) , P ( Y$_{0}$ ≤ y$_{0}$, Y ≤ y$_{1}$ | 1 , z ) ≥ ( 1 p ( z )$^{)}$$^{2}$ P ( Y$_{0}$ ≤ y$_{0}$ | 1 , z ) P ( Y$_{1}$ ≤ y$_{1}$ | 1 , z ) . F $^{L}$( y$_{0}$, y$_{1}$ ) = sup z ∈ Ξ $^{{}$P ( y$_{0}$ | 0 , z ) L wst 10 ( y$_{1}$, z ) + L wst 01 ( y$_{0}$, z ) P ( y$_{1}$ | 1 , z ) $^{}}$, F U ( y$_{0}$, y$_{1}$ ) = inf z ∈ Ξ    min $^{{}$P ( y$_{0}$ | 0 , z ) (1 − p ( z )) , U wst 10 ( y, z ) } + min $^{{}$U wst 01 ( y$_{0}$, z ) , P ( y$_{1}$ | 1 , z ) p ( z ) }   $_{}$. In Subsection \ ref { c2s3s3 } , I proved that Also by (8) and (9), for any z ∈ Ξ , Proof. The proof of Theorem 2 consists of two parts: sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ and sharp bounds on the DTE under M. 1 − M. 5 and CPQD.