Consider the binary symmetric channel described in Figure. Let Po denote the prob. ability of sending binary symbol x₀ = 0, and let p₁ = 1 ? P₀ denote the probability of sending binary symbol x₁ = 1, Let p denote the transition probability of the channel

(a) Show that the mutual information between the channel input and channel output is given by

(b) Show that the value of p₀ that maximizes I{X; Y) is equal to 1/2.

(c) Hence, show that the channel capacity equals. C = 1 ? H(p)

I(X; Y) = H(2) H(p) %3D H(2) = z log::) + (1 - 2) log.() z = Pop + (1 - Po)l(1 - p) Hp) = p log:) + (1 - p) log.) 1-p *; = 1 o 1-P