Question 10In this question we explore the application of MAP and LMS estimation in a communication channel.Suppose we want to send a signal X, but this is corrupted by additive noise N such that the signalat the receiver is Y, whereY = X+ N.We assume the noise is Gaussian with zero mean and a known variance, i.e.,N ~ N(0, 0²).We further assume that X is a continuous random variable with half of the probability clustered at+1 and half of the probability clustered at –1. (We say this, instead of calling X a discrete randomvariable, is so that the estimate does not necessarily have to be ±1. We can formalize it in more rigorousmathematics using a delta function, but that is beyond the scope of this class.)1. Find the MAP estimate of X.2. Find the LMS estimate of X.

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Answered: Question 10 In this question we explore…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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3. Let X1, X2, ..., Xn be a random sample from the population N(u, o²) and Y1, Y2,...,Ym a random sample from the population N(uy, o²), where both means (ua and uy) are assumed known. Note that the two distributions have a common variance of o?. (a) If the X's and Y's are independent, show that the MLE for the common variance is: E(Xi - Ha)² +E(Y; - Hy)² n + m (b) Is the MLE found above unbiased for o??
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QUESTION 6 Suppose we know that X and Y are random variables with var(X) = 2, var(Y) = 3, and cov(X,Y) = -1. Find var (2X – Y). Round to 4 decimal places if needed.
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Suppose we obtain n observations from a sinusoid contaminated by Gaussian noise: X = A· sin(2 · T· f·k+¢) + Wk, k = 1, 2, ...,n, %3D where Wg are independent identically distributed (iid) Gaussian random variables with mean o and variance o?. The frequency f and phase o are known constants, while A is an unknown constant. Find the maximum likelihood estimator for A. Show all work.
In this problem, we consider target detection in a radar system. Upon observing a realization a of the random variable X which represents the return signal in noise, a decision has to be made as to whether the target is present or absent. The target amplitude is known and equal to A > 0. The observation is made in zero-mean Gaussian noise W whose variance is known and equal to 2. If the target is present (hypothesis H1), X = A+W. If the target is absent (hypothesis Ho), X = W . The likelihood ratio test for this problem is stated as follows: If fi (x)/fo(x) > n= decide in favor of H1; otherwise, decide in favor of Ho.lf n = 1, determine the probability of correct detection (target declared present when indeed present) and the probability of false alarm (target declared present when in fact absent). Express these probabilities in terms of the ratio a = A/o. If a = 4, which of the following represents the correct probabilities of false alarm and correct detection, respectively? O None of…
2. Review on estimators. Let Y = (Y1, 2, 3, 4) be a random sample from the normal distribution N(μ, 2²), where the expected value of the normal distribution μER is unknown, and the variance parameter has four (4) as a value. We examine the following parameter μ estimator 1 ₪ (Y) = ± ± (Y₁ + Y₂ + Y3 + 2Y₁) 4 Calculate the bias of the estimator (Y). Is the estimator unbiased? Calculate the variance and the mean squared error (MSE) of the estimator. The problems deal with the confidence intervals (Chapter 5 topics) and especially testing theory (Chapter 6).
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Suppose that Y₁ is a negative binomial random variable with p= .75 and r = 6, Y₂ is a random variable with mean -5 and variance 11, and Y3 is a gamma random variable with 1.6 and a = 5.9. Further suppose that all of these random variables are independent. (a) What is E[2Y₁ − 6Y2 + 5Y₁Y3]? (b) What is the variance of 3Y₁ +5Y2 - 8Y3? (c) What is the mgf of 7Y₁ +4Y3? 0 =

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