Solve all the parts.

Problem $1$ In this problem, we will empirically verify the ROC curve and the Bayes risk. Consider the

following hypothesis problem: $H_0$:$YN(0,1)$ versus $H_1$:$YN(m,1).$ Assume uniform costs and the

priors are given by $P_0$ and $P_1.$

$($a$)$ Compute the Bayes test. Also specify the expressions of $P_D,P_F$ and the Bayes risk $J$ in terms of the SNR

parameter $d.$

$($b$)$ Consider three different values of $P_0=\{0\mathrm{.}1,0\mathrm{.}25,0\mathrm{.}4\}$ and different values of $d=\{0,0\mathrm{.}1,0\mathrm{.}2,$cdots,$2\}.$ For

each value of $P_0$ and $d,$ generate ${10}^6$ independent random observation points $Y$ according to the appropriate

prior and conditional distributions $($you can use Matlab functions for this$).$ Perform the Bayes test on the

observations and compute the empirical Bayes risk by averaging the risks over all the observations.

$($c$)$ Plot the theoretical Bayes risk as a continuous function of din$[0,2].$ In the same figure, mark discrete

points $($in different color$)$ corresponding to the empirical Bayes risk for $d=\{0,0\mathrm{.}1,0\mathrm{.}2,$cdots,$2\}.$ Repeat the

same exercise in the same figure for all three values of $P_0=\{0\mathrm{.}1,0\mathrm{.}25,0\mathrm{.}4\}.$ Comment on what you observe.

$($d$)$ For each value of $d=\{1,2,3\},$ plot the theoretical ROC curve. On the same curve, mark discrete points

$($in different color$)$ corresponding to empirical values of $P_D$ and $P_F$ for each value of $P_0=\{0\mathrm{.}1,0\mathrm{.}25,0\mathrm{.}4\}.$

Comment on what you observe.