A binary communications system accepts $\Theta$, which is " 0 " or " 1 ", as input and outputs $X$, " 0 " or " 1 ", with probability of error $P[\Theta \neq X]=p=10^{-3}$. Suppose the sender uses a repetition code whereby each " 0 " or " 1 " is transmitted $n$ independent times, and the receiver makes its decision based on the $n=8$ corresponding outputs. Assume that $1 / 5=P[\Theta=1]=\alpha=1-P[\Theta=0]$.
(a) Find the maximum likelihood decision rule that selects the input which is more likely for the given $n$ outputs. Find the probability of Type I and Type II errors, as well as the overall probability of error $P_c$.
(b) Find the Bayes decision rule that minimizes the probability of error. Find the probability of Type 1 and Type II errors, as well as $P_e$.
(c) For the decision rules in parts a and b find $n$ so that $P_e=10^{-9}$.