content_image/299635.jpg content_image/299634.jpg content_image/299633.jpg content_image/299632.jpg
<img src='content_image/299635.jpg'>

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. 

## Part 1. Sharp bounds on the joint distribution of Y$_{0}$ and Y$_{1}$ 

In Subsection \ ref { c2s3s3 } , I proved that 

Also by (8) and (9), for any z ∈ Ξ , 

<img src='content_image/299634.jpg'>

<img src='content_image/299633.jpg'>

Finally, the lower bound P ( Y$_{0}$ ≤ y$_{0}$, Y$_{1}$ ≤ y$_{1}$ ) can be obtained by taking the intersection over all z ∈ Ξ , 

<img src='content_image/299632.jpg'>

