A widely used digital radio system transmits pairs of bits at a time. The input to the system is a pair $\left(\Theta_1, \Theta_2\right)$ where $\Theta_i$ can be +1 or -1 and where the output of the channel is a pair of independent Gaussian random variables $(X, Y)$ with variance $\sigma^2$ and means $\Theta_1$ and $\Theta_2$, respectively. Assume $P\left[\Theta_i=1\right]=\alpha=1-P\left[\Theta_i=0\right]$ and that the input bits are independent of each other. The receiver observes the pair $(X, Y)$ and based on their values decides on the input pair $\left(\Theta_1, \Theta_2\right)$.
(a) P!ot $f_{X, Y}\left(x, y \mid \Theta_1, \Theta_2\right)$ for the four possible input pairs.
(b) Let the cost be zero if the receiver correctly identifies the input pair, and let the cost be one otherwise. Show that the Bayes' decision rule selects the input pair $\left(\theta_1, \theta_2\right)$ that maximizes:
$$
f_{X, Y}\left(x, y \mid \theta_1, \theta_2\right) P\left[\Theta_1,=\theta_1, \Theta_2=\theta_2\right] .
$$
(c) Find the four decision regions in the plane when the inputs are equally likely. Show that this corresponds to the maximum likelihood decision rule.