Question

Consider a binary hypothesis testing where the observation Y has the following PDFs $\qquad f(y|H_0) = \begin{cases} A_0(a - |y|), & \text{for } |y| \le a, \\ 0, & \text{for } |y| > a, \end{cases} \text{ given } H_0,$ $\qquad f(y|H_1) = \begin{cases} A_1(b - |y|), & \text{for } |y| \le b, \\ 0, & \text{for } |y| > b, \end{cases} \text{ given } H_1.$ Let's assume $b > a > 0$. a) Calculate the constants $A_0$ and $A_1$ in terms of parameters $a$ and $b$. b) Assume uniform cost assignment $c_{ij} = 1 - \delta_{ij}$ and find the optimum decision rule based on Bayes criterion. c) Evaluate $P_D$ and $P_{FA}$ as a function of the detection threshold $\Lambda_0 = \frac{\pi_0}{\pi_1}$. Assume $b = 2a$ and sketch the ROC curve.

          Consider a binary hypothesis testing where the observation Y has the following PDFs
$\qquad f(y|H_0) = \begin{cases} A_0(a - |y|), & \text{for } |y| \le a, \\ 0, & \text{for } |y| > a, \end{cases} \text{ given } H_0,$ 
$\qquad f(y|H_1) = \begin{cases} A_1(b - |y|), & \text{for } |y| \le b, \\ 0, & \text{for } |y| > b, \end{cases} \text{ given } H_1.$ 
Let's assume $b > a > 0$.
a) Calculate the constants $A_0$ and $A_1$ in terms of parameters $a$ and $b$.
b) Assume uniform cost assignment $c_{ij} = 1 - \delta_{ij}$ and find the optimum decision rule based on Bayes
criterion.
c) Evaluate $P_D$ and $P_{FA}$ as a function of the detection threshold $\Lambda_0 = \frac{\pi_0}{\pi_1}$. Assume $b = 2a$ and sketch
the ROC curve.

Consider a binary hypothesis testing where the observation Y has the following PDFs
f(y|H0) = A0(a - |y|), for |y| ≤ a,
0, for |y| > a, given H0,
f(y|H1) = A1(b - |y|), for |y| ≤ b,
0, for |y| > b, given H1.
Let's assume b > a > 0.
a) Calculate the constants A0 and A1 in terms of parameters a and b.
b) Assume uniform cost assignment cij = 1 - δij and find the optimum decision rule based on Bayes
criterion.
c) Evaluate PD and PFA as a function of the detection threshold Λ0 = (π0)/(π1). Assume b = 2a and sketch
the ROC curve.

Added by Alejandro S.

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