Using a certain binary digital communication system receiver produces a decision variable D = A + N, where A can be 1 or -1 and N has a uniform distribution on [-1.5, 1.5], yielding the conditional density functions
$f_D(x \mid A = 1) = \begin{cases} \frac{1}{3}, & -0.5 < x < 2.5; \\ 0, & \text{otherwise.} \end{cases}$
$f_D(x \mid A = -1) = \begin{cases} \frac{1}{3}, & -2.5 < x < 0.5; \\ 0, & \text{otherwise.} \end{cases}$
Find the probability of an error using the optimum decision rule assuming the two alternatives for A are equally likely.
D = A + N
