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## Theorem 1 

Theorem 1 Under M. 1 − M. 4 , sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ , their joint distribution and the DTE are obtained as follows: for d ∈ { 0 , 1 } , y ∈$_{R}$ , δ ∈$_{R}$ , and ( y$_{0}$, y$_{1}$ ) ∈$_{R}$ ×$_{R}$ , 

where 

<img src='content_image/315807.jpg'>

<img src='content_image/315808.jpg'>

Proof. The proof consists of three parts: sharp bounds on (i) marginal distributions, (ii) the joint distribution, and (iii) the DTE. 

## Part 1. Sharp bounds on marginal distributions F$_{0}$ ( · ) and F$_{1}$ ( · ) 

