ps2 - Massachusetts Institute of Technology Department of...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.432 Stochastic Processes, Detection and Estimation Problem Set 2 Spring 2004 Issued: Tuesday, February 10, 2004 Due: Thursday, February 19, 2004 Reading: For this problem set: Chapter 2, through Section 2.5.1 Next: Chapter 2, Chapter 3 through Section 3.2.4 Problem 2.1 Let x and y be statistically independent random variables with probability density functions 1 1 p x ( x ) = λ ( x + 1) + λ ( x 1) 2 2 and 1 y 2 p y ( y ) = exp , 2 �δ 2 2 δ 2 and let z = x + y , and w = xy . (a) Find p z ( z ), the probability density function for z . (b) Find the conditional probability density functions p z | x ( z x = 1) and p z | x ( z x = | | 1). (c) Find the mean values x ¯ = m x and y ¯ = m y , the variances δ y 2 and δ 2 , and the w covariance yw . Are y and w uncorrelated random variables? Are y and w statistically independent random variables? (d) Are y and w Gaussian random variables? Are they jointly Gaussian? Explain. Problem 2.2 Let x 1 and x 2 be zero-mean jointly Gaussian random variables with covariance matrix 34 12 x = . 12 41 (a) Verify that x is a valid covariance matrix. (b) Find the marginal probability density for x 1 . Find the probability density for y = 2 x 1 + x 2 . 1
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(c) Find a linear transformation defining two new variables x x 1 2 = P x 1 x 2 so that x 1 and x 2 are statistically independent and so that PP T = I , where I is the 2 × 2 identity matrix. Problem 2.3 (practice) Let x be an N -dimensional zero-mean random vector whose covariance matrix has eigenvalues 1 > 2 > > N , · · · and corresponding eigenvectors , π N . π 1 , π 2 , · · · Suppose we wish to approximate x as a scalar random variable b times
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