$2.$ Using the same data in the Table $1,$ now we test a list of threshold used to make final positive $/$negative calls. Threshold list is

$\backslash [$

$[0\mathrm{.}4,0\mathrm{.}50,0\mathrm{.}55,0\mathrm{.}6,0\mathrm{.}65,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9],$

$\backslash ]$

a$.$ please find TP$,$ FP$,$ TN$,$ FN$,$ and Sensitivity $($TP Rate: TPR$),$ Specificity $($TN Rate: TNR$),$ and False Positive$($FPR: $1-$TNR$),$ fill in the table below.

b$.$ Draw a ROC curve for this classifier $($Receiver Operating Characteristic curve$)$; a graph showing the performance of a classification model at all classification thresholds with X axis being FPR $(1-$specficity$),$ and Y axis being TPR $($sensitivity$)$ ; Note the redline is a reference line.

c$.$ AUC: Area Under the ROC Curve is a useful metric to evaluate the classification model. It measures the entire two$-$dimensional area underneath the entire ROC curve $($think like integral calculus from $(0,0)$ to $(1,1).$ What is the maximum value of AUC possibly? What is the AUC value for the classifier above, and explain why AUC is a good measure for classifier.