Recall the example (2): for d ∈ { 0 , 1 } , where ( ν$_{0}$, ν$_{1}$ ) ⊥ U. CPQD requires that ν$_{0}$ and ν$_{1}$ be positively quadrant dependent. As a restric- tion on dependence between ε$_{0}$ and ε$_{1}$ conditional on U, CPQD has some information on the joint distribution of Y$_{0}$ and Y$_{1}$ , but not marginal distribution of Y$_{d}$, which is identified by the distribution of ε$_{d}$ conditional on U for d ∈ { 0 , 1 } . Specifically, the lower bound on the conditional joint distribu- tion of ε$_{0}$ and ε$_{1}$ given U improves under CPQD as shown in (6). This is due to the nonnegative sign restriction on dependence between ε$_{0}$ and ε$_{1}$ given U implied by CPQD. Without CPQD, the sharp lower bound and the upper bound on the conditional joint distribution are achieved when the con- ditional distributions of ε$_{0}$ given U and ε$_{1}$ given U are perfectly negatively dependent and perfectly positively dependent, respectively. Under CPQD, however, the dependence is restricted to range from independence to perfectly positive dependence without any negative dependence. Therefore, the lower bound under CPQD is attained when their conditional dependence is independent. I show that the lower bound on the unconditional joint distribution can be improved from the improved lower bound on the conditional joint distribution. Chebyshev’s integral inequality is useful for deriving the improved lower bound on the joint distribution of Y$_{0}$ and Y$_{1}$ under CPQD: Chebyshev’s Integral Inequality If f and g : [ a, b ] −→$_{R}$ are two comonotonic functions, then To establish bounds on the joint distribution, recall (5). For e$_{0}$ = m − $^{1}$(0 , y$_{0}$ ) and e$_{1}$ = m − $^{1}$(1 , y$_{1}$ ) for ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , Now I require the additional assumption: 17 P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ | 0 , z ) = P ( ε$_{0}$ ≤ e$_{0}$, ε$_{1}$ ≤ e$_{1}$ | U > p ( z )) . 1 b − a b ∫ a f ( x ) g ( x ) dx ≥   1 b − a b ∫ a f ( x ) dx$^{}$    1 b − a b ∫ a g ( x ) dx$^{}$ $_{}$. ε$_{d}$ = ρ$_{d}$U + ν$_{d}$,