Multi-user Information Theory 2. Definition 1 (physically Degraded Bc) Definition 2 (stochastic Degraded Bc)

A broadcast channel given by p(y 1 , y 2 |x) is said to be (physically degraded) if p(y 1 , y 2 |x) = p(y 1 |x)p(y 2 |y 1), ∀x ∈ X , ∀y 1 ∈ Y 1 , ∀y 2 ∈ Y 2. A broadcast channel given by p(y 1 , y 2 |x) is said to be (stochastic degraded) if its conditional marginal distributions p(y 1 |x) and p(y 2 |x) are the same as that of a physically degraded broadcast channel; that is, if there exists a distribution p ′ (y 2 |y 1) such that p(y 2 |x) = y1 p(y 1 |x)p ′ (y 2 |y 1). Theorem 1 : The capacity region of a broadcast channel depends only on the conditional marginal distributions p(y 1 |x) and p(y 2 |x).