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University of Texas - EE 351Khw10_spring2016_sol-1

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EE 351K Probability, Statistics and Stochastic Processes Homework 10 Spring 2016 Topics: Classical Statistical Inference: MAP and ML estimation,LMS and Linear LMS estimation Q. 1 At a computer disk drive factory, inspectors randomly pick a product from production lines to detect a failure. If the production lines are normal, this failure rate q0 = 10−4. However occasionally some problems occur in the lines, in which case the rate goes up to q1 = 10−1. Let Hi denote the hypothesis that the failure rate is qi. Every morning, an inspector chooses drives at random from the previous day’s production and tests them. If a failure occurs too soon, the company stops production and checks the critical part of the process. Production line problems occur at random once every 10 days, so that P(H1) = 0:1 = 1−P(H0). 1. Based on N, the number of drives tested up to and including the first failure, design a MAP test. 2. Calculate the probability of ‘false alarm’ (i.e. our MAP test computed in previous part concludes that the rate is q1 wrongly) and the probability of ‘missed detection’ (i.e. our MAP test fails on detect that the rate is q1). 3. Based on this, calculate the probability of detection error Pe. Sol: 1. Given a failure rate of qi, N is a geometric RV with expectation 1=qi. Thus PNjHi(n) = qi(1 − qi)n−1. Thus MAP test corresponds to PNjH0(n) PNjH1(n) ≥ P(H1) P(H0) for hypothesis H0. This in turn implies n ≥ n∗ = 1+ log q1P(H1) q0P(H0) log 1 1− −q q0 1 for hypothesis H0 and this yields n∗ ≈ 45:8. Thus we see n ≥ 46 is the decision for hypothesis H0. Then the inspector tests at most 45 drives in order to reach a conclusion about the failure rate. If the first failure occurs before test 46, the company assumes that the failure rate is 10−1. If the first 45 drives passes the test, then N ≥ 46 and company assumes that failure rate is 10−4. 2. The error probabilities of false alarm (PFA) and missed detection (PMISS) are: PFA = P(N ≤ 45 j H0) = 1−(1−10−4)45 = 0:0045; PMISS = P(N > 45 j H1) = (1−10−1)45 = 0:0087; 3. The total probability of error is Pe = P(H0)PFA +P(H1)PMISS = 0:0049. Q. 2 Romeo and Juliet start dating, but Juliet will be late on any date by a random amount X, uniformly distributed over the interval [0, q]. The parameter q is unknown and is modeled as the value of a random variable Q, uniformly distributed between zero and one hour. 1. Assuming that Juliet was late by an amount x on their first date, how should Romeo use this information to compute the a posteriori distribution of Q? 2. Find the MAP estimate of Q based on the observation X = x: 3. Find the Least Mean Square estimate of Q based on the observation X = x: 4. Derive the Linear Least Mean Square estimator of Q based on X: Sol:

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