EECS 564________________________PROBLEM SET #1________________________Winter 1999

ASSIGNED: January 08, 1999
DUE DATE: January 15, 1999

READ: Chapter 2 of Srinath (skip Section 2.5 for now), and also Sections 3.1-3.3.
Pages 8-11 on linear systems will be crucial for the Kalman filter; read carefully.
THIS WEEK: Basic hypothesis testing using Bayesian and Neyman-Pearson criteria.

1. Gaussian random variable x, with mean m and variance s2, is passed through either:
H0: y=x2 or H1: y=ex. We observe y. Compute the LRT; DON'T try to simplify it.
2. We observe K independent and identically distributed 0-mean Gaussian RVs: r1...rK.
Under H0 their variance is s02; under H1 their variance is s12.
1. Compute the likelihood ratio test (any threshold). Simplify it as much as possible.
2. Show that R12+...+RK2 is a sufficient statistic.
3. Compute expressions for PF and PM=1-PD.
Express your answer in terms of the PDF for a chi-square distribution (see p. 57).
4. Plot the ROC for K=1 (what happens to the chi-square?), s02=1, s12=2.

3. We receive a shipment of N widgets, all of which came from either:
• Batch #0, with probability p0, for which Pr[defective widget]=q0, or
• Batch #1, with probability p1, for which Pr[defective widget]=q1> q0.
We test the N widgets and find that y of them are defective.
We wish to determine whether the widgets all came from batch #0 or batch #1, based on y.
1. Compute the MEP (minimum error probability) decision rule.
2. Plot the ROC for the LRT for this problem when N=2.

4. In honor of the upcoming "Star Wars" prequel:
Luke has lost his droid. To find it, he fires a ladar* pulse at a location.
The return y from the ladar pulse is exponentially distributed (Le-LY,Y> 0), with
L=3 if the droid is absent, and L=1 if the droid is present.
Design a Neyman-Pearson test with level of significance=0.01, and compute the power PD.
*DARTH LADAR=Don and Ron's Transistorized Handheld Laser Detection and Ranging.