Because of differences in printings and a typo, here is the version of Problem 2.41 you should solve:

41. Consider two one-dimensional triangle distributions having different means but the same width:

p(x|ω) = T(μ_i,δ) = { (δ - |x-μ_i|)/δ² for |x- μ_i| < δ

0 otherwise

with μ₂ > μ₁. We define a new discriminability here as d_T^' = (μ₂ - μ₁)/δ .

a) Write an analytic function, parameterized by d_T^' , for the operating characteristic curves in each of the relevant ranges: μ₂ - μ₁ > 2, 2 > μ₂ - μ₁ > 1, etc.

b) Plot these novel operating characteristic curves for d_T^' = {0.0, 0.2, 0.4, ..., 2.0}. Interpret your answer for the case d_T^' = 2.0.

c) Suppose we measure Pr[x > x*| x∈ ω₂] = 0.6 and Pr[x > x*| x∈ ω₁] = 0.2. What is d_T^'? What is the Bayes error rate?

d) Infer the decision rule implied in part (c). That is, express x* in terms of the variables given in the problem.

e) Suppose we measure Pr[x > x*| x∈ ω₂] = 0.9 and Pr[x > x*| x∈ ω₁] = 0.1. What is d_T^'? What is the Bayes error rate?

f) Infer the decision rule implied in part (e). That is, express x* in terms of the variables given in the problem.