Q$2.$ Classification Evaluation Measures $($Total $6$pt$)$

Consider a test data of $1000$ samples with two classes: $+$ class $(100$ samples$)$ and class $(900$ samples$).$ We have two random classifiers C$1$ and C$2.$

We note that classifier C $1$ classifies test data to $+$ class randomly with a probability p $\backslash ((0\backslash$leq p $\backslash$leq $1)\backslash )$ and classifier C $2$ classifies test data to $+$ class randomly with a probability $2$ p$.$

$($a$)(3$pt$)$ What is the expected TPR and FPR for C $1$ and C$2?$

$($b$)(3$pt$)$ Is C$2$ a better classifier than C$1?$ Hint: The random guess line in an ROC curve corresponds to $\backslash ($ T P R$=$F P R $\backslash ).$