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.

- Gaussian random variable x, with mean m and variance s
^{2}, is passed through either:

H_{0}: y=x^{2}or H_{1}: y=e^{x}. We observe y. Compute the LRT; DON'T try to simplify it.

- We observe K independent and identically distributed 0-mean Gaussian RVs: r
_{1}...r_{K}.

- Under H
- Compute the likelihood ratio test (any threshold). Simplify it as much as possible.
- Show that R
_{1}^{2}+...+R_{K}^{2}is a sufficient statistic. - Compute expressions for P
_{F}and P_{M}=1-P_{D}.

Express your answer in terms of the PDF for a chi-square distribution (see p. 57). - Plot the ROC for K=1 (what happens to the chi-square?), s
_{0}^{2}=1, s_{1}^{2}=2.

_{0}their variance is s_{0}^{2}; under H_{1}their variance is s_{1}^{2}.

- We receive a shipment of N widgets, all of which came from either:
- Batch #0, with probability p
_{0}, for which Pr[defective widget]=q_{0}, or - Batch #1, with probability p
_{1}, for which Pr[defective widget]=q_{1}> q_{0}.

We wish to determine whether the widgets all came from batch #0 or batch #1, based on y.- Compute the MEP (minimum error probability) decision rule.
- Plot the ROC for the LRT for this problem when N=2.

- Batch #0, with probability p
- 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 P_{D}.

^{*}DARTH LADAR=Don and Ron's Transistorized Handheld Laser Detection and Ranging.