In some applications, a classifier is allowed to 'reject' a test example rather than classifying it. Consider a case in which the coast of misclassification is $10 but the cost of having a human manually make the decision is only $3. Suppose our problem is a two class problem, so either C=0 or C=1, and P(C=0|x) = 1 - P(C=1|x). So for our classifier, the possible actions are:

a0	Classify as class 0 (C=0)
a1	Classify as class 1 (C=1)
a2	Reject (send to human)

and losses are:

	Loss ($)	Description
00	0	Classify as class 0 when the true class is also 0
11	0	Classify as class 1 when the true class is also 1
01	10	Classify as class 0 when the true class is 1
10	10	Classify as class 1 when the true class is 0
20	3	Reject when the true class is 1
21	3	Reject when the true class is 0

a. What is the expected risk for each action? (Meaning: write a formula for R(a_i | x) in terms of P(C=1|x) for each of the three actions.)

b. What is the optimal decision rule? (Meaning : write something of the form: choose a₀ if .... , chose a₁ if ... otherwise reject.)

c. Suppose P(C=1|x) = 0.2 for some input x, what should the classifier's action be?

d. Suppose P(C=1|x) = 0.4 for some input x, what should the classifier's action be?