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 . L

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. 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, θ]. The parameter θ is unknown and is modeled as the value of a random variable Θ, 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 Θ? 2. Find the MAP estimate of Θ based on the observation X = x. 3. Find the Least Mean Square estimate of Θ based on the observation X = x. 4. Derive the Linear Least Mean Square estimator of Θ based on X. Q. 3 Let X be a random variable uniform over the interval [2,12]. Suppose we observe X with some random error N which is uniformly distributed over interval [0,1] and independent of X. i.e., Y = X +N 1. Find the Least mean square estimate of X based on the observation Y = y. 2. Find the mean square error. Q. 4 A discrete-time sequence s[n] is transmitted over a noisy channel and retrieved. The received sequence x[n] is modeled as x[n] = a×θ[n]+w[n] where w[n] and a represents the channel noise and attenuation respectively. At a particular time instant n = n0, suppose x[n0], θ[n0] and w[n0] are random variables, which we denote as X, Θ and W respectively. We assume that Θ and W are independent, that W is distributed as a Gaussian N(0,1) and the signal Θ is distributed as a Gaussian N(10,3). Please compute: 1. The LMS estimator of θ given X in function of a. 2. The Linear LMS estimator of θ given X. How are both estimators related? Q. 5 The amplitude of a random non-negative d.c. signal is described by a random variable X, with an exponential density function with parameter µ > 0. In other words, for any x ≥ 0, fX (x) = µe−µx This random signal is fed as an input to a non-linear squaring system, whose output amplitude is described by the random variable Y = X 2 . 1 1. Determine an expression for the pdf of Y. 2. Determine the conditional expectation of X given Y, i.e., E[X|Y]. and provide an expression for the minimum mean square estimator (MMSE) of X given Y. 3. Determine an expression for a linear estimator (of the form Xˆ = aY +b for suitable real values of a and b) which minimizes the mean-square error between Xˆ and X. In other words, determine the values of a,b that minimizes E[(X − aY − b) 2 ]. You may use the following formula: For any n ≥ 0, and c > 0, Z ∞ 0 x n e −cxdx = n! c n+1 Q. 6 The number θ of shopping carts in a store is modeled as the value of a random variable Θ, uniformly distributed between 1 and 100. Carts are sequentially numbered between 1 and θ. You enter the store, observe the number X on the first cart you encounter, assumed uniformly distributed over the range 1,…,θ, and use this information to estimate θ. Please find: 1. The MAP estimator of θ given X 2. The LMS estimator of θ given X 3. The LLMS estimator of θ given X. 2

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