Problem 1.

Find the Maximum Likelihood (ML) test for a single observation x(0) to distinguish between H₀ and H₁ if

H₀ : P_x(0)(x(0)) = 1/2exp (-|x(0)|)

H₁ : P_x(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 P_D while PFA = 1/2e.

3. What is P_D 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 P_FA for the optimal Neyman-Pearson detector and sketch the ROC.

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.