Problem 1.
Find the Maximum Likelihood (ML) test for a single observation x(0) to distinguish between H0 and H1 if
H0 : Px(0)(x(0)) = 1/2exp (-|x(0)|)
H1 : Px(0)(x(0)) = 1/2exp (-|x(0) - A|)
when A > 0. Find the error probability. Find the probability of deciding H1 when H1 is in fact true.
Problem 2.
For the previous problem, find the Neyman-Pearson (NP) test (to distinguish between H0 and H1) that achieves PFA = 0.1. What is PD? Does your test depend on the value of A? Explain.
Problem 3.
We make a single observation x(0) with
H0 : x(0) = ω(0)
H1 : x(0) = 6 + ω(0)
with
pω(0)(ω(0)) = 1/2exp (|ω(0)|)
This is an instance of a signal (6) in additive noise (ω(0)), and we wish to detect the signal while avoiding false alarms.
1. Find the log likelihood ratio as a function of x(0).
2. Determine the test that maximizes PD while PFA = 1/2e.
3. What is PD for your test?
Problem 4.
The PDFs of a single observation x(0) under H0 and H1 are shown below. Find PD as a function of PFA for the optimal Neyman-Pearson detector and sketch the ROC.
![2244_Figure2.jpg](https://secure.tutorsglobe.com/CMSImages/2244_Figure2.jpg)
Problem 5.
You measure iid x(n), n = 0, 1, ....., N - 1.
H0: px(n)(x(n)) = exp(-x(n)), if x(n) > 0
0, otherwise.
and
H1: px(n)(x(n)) = 1/µ exp(-x(n)/µ), if x(n) > 0
0, otherwise.
1. Find the log likelihood ratio as a function of x(0).
2. Determine the test that maximizes PD while PFA.
3. What is PD for your test?
Example.
You know a single observation x(0) is distributed under two hypothesis as follows:
H0: px(0)(x(0)) = 1/2, if x ∈ {-1, 1}
0, otherwise.
and
H1: px(0)(x(0)) = 3/2x2 if x ∈ {-1, 1}
0, otherwise.
The probability of occurrence of H0 and H1 are Pr(H0) = 0.95 and Pr(H1) = 0.05.
1. Determine a detector based on x(0) to minimize the error probability.
2. Now suppose that the cost of deciding H0 when H1 is true is $475 and the cost of deciding H1 when H0 is true $48. Under these assumption about the cost, determine a decision rule based on x(0) that minimizes the expected cost.
3. For your decision rules in (1) and (2), calculate the probability of decisding H1 when H1 is in fact true.