Now to derive the lower bound F $^{L}$( y$_{0}$, y$_{1}$ ) , let G ( y ) = F$_{1}$ ( y ) − ( F$_{0}$ ( y ) − F$_{0}$ ( y$_{0}$ )) $^{+}$− ( F$_{1}$ ( y ) − F$_{1}$ ( y$_{1}$ )) $^{+}$. Then for y$_{0}$ < y$_{1}$$_{,}$ and so, Since F$_{1}$ ( y$_{1}$ ) − F$_{0}$ ( y$_{1}$ ) + F$_{0}$ ( y$_{0}$ ) ≥ max (0 , F$_{0}$ ( y$_{0}$ ) + F$_{1}$ ( y$_{1}$ ) − 1) , for y$_{0}$ < y$_{1}$, Corollary 2 (Bounds on the marginal distributions of potential outcomes) Under M. 1 − M. 4 , PSM and MTR, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ , their joint distribution and the Corollary 2 G ( y ) =        F$_{0}$ ( y$_{0}$ ) + F$_{1}$ ( y$_{1}$ ) − F$_{0}$ ( y ) , if y$_{1}$ ≤ y F$_{1}$ ( y ) − F$_{0}$ ( y ) + F$_{0}$ ( y$_{0}$ ) , if y$_{0}$ ≤ y < y$_{1}$ F$_{1}$ ( y ) , if y < y$_{0}$ . sup y ∈$_{R}$ G ( y ) = sup y$_{0}$ ≤ y ≤ y$_{1}$ { F$_{1}$ ( y ) − F$_{0}$ ( y ) + F$_{0}$ ( y$_{0}$ ) } F $^{L}$( y$_{0}$, y$_{1}$ ) = sup y$_{0}$ ≤ y ≤ y$_{1}$ { F$_{1}$ ( y ) − F$_{0}$ ( y ) + F$_{0}$ ( y$_{0}$ ) } . 45