Problem $5$: Consider the evaluation and plotting of a Receiver Operating Characteristics $($ROC$)$ for the

simple binary detection problem where

$=P_D=P_{FA}(x_c)(x_c)x_c{R}_1=\{x$:$x{x}_c\}\{$:$R_0=\{x$:$x{x}_c\}.)=0\mathrm{.}001,0\mathrm{.}01,0\mathrm{.}1x_{c}0\mathrm{.}00010\mathrm{.}0010\mathrm{.}010\mathrm{.}1$