Solve the following problem:
The channel C (known as the Z channel) is shown in Figure .
1. Find the input probability distribution that achieves capacity.
2. What is the input distribution and capacity for the special cases ε = 0, ε = 1, and ε = 0.57?
3. Show that if n such channels are cascaded, the resulting channel will be equivalent to a Z channel with ε1 = εn.
4. What is the capacity of the equivalent Z channel when n → ∞?