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

following hypothesis problem: H$0$ : Y $$ N$(0,1)$ versus H$1$ : Y $$ N$($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 PD$,$ PF 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 $=\{0,0\mathrm{.}1,0\mathrm{.}2,,2\}.$ For

each value of P$0$ and d$,$ generate $106$

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 d in $[0,2].$ In the same figure, mark discrete

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

$1$

$($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 PD and PF for each value of P$0=\{0\mathrm{.}1,0\mathrm{.}25,0\mathrm{.}4\}.$

Comment on what you observe.

$($e$)$ You will have to also submit the Matlab code file. The file should run without errors and generate the two

plots in parts $($c$)$ and $($d$).$ Properly label all the axes, the title of the plot and the legend of the plot. Include

these two figures in the pdf file as well