$(24$ points$)$ Consider the following sinusoidal carrier wave, X$($t$),$ with a carrier frequency of fc and a carrier phase $($t$)$: X$($t$)=$ cos$(2$fct $+($t$))$ The phase of this carrier is altered or modulated to encode a stream of binary data to be transmitted. Assume that the data stream is an I.I.D sequence of RVs that are distributed Bernoulli$($p$).$ Consider the nth bit $($bn in $\{0,1\})$ in the stream of binary data is assigned to $/2$ with probability p and $$$/2$ with probability $(1$ p$)$: