EE562a_HW_1

EE562a_HW_1 - Homework 1 Due Monday June 5, 2006 Work the...

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EE 562a Homework 1 Due Monday June 5, 2006 Work the following 11 problems . Problem 1. Suppose the random variable X has density f ( x )= ( 2 e - 2 x ,x> 0 0 , elsewhere. a. Compute P ( X< 1). b. Let Y = X 2 +1. Find E ( Y ). Problem 2. a. Suppose X is exponentially distributed with parameter λ , i.e., f ( x ( λe - λx 0 0 , elsewhere. Derive the moment generating function for X . b. Suppose the random variable Y has moment generating function M Y ( s )=(1 2 s ) - 1 ,s< 1 2 . Find the mean and variance of Y . Problem 3. Let X and Y be independent and normally (Gaussian) dis- tributed random variables each with mean 0 and variance σ 2 . De±ne U = X 2 + Y 2 and V = X X 2 + Y 2 . Show U and V are independent. Problem 4. Compute (by hand) the eigenvalues and eigenvectors for the matrix A = 3 10 12 1 0 13 . 1
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Remark: For most homework problems in this class you may use Matlab when applicable but for this problem you should do the calculations by hand. Problem 5. Let Y ( u )= HX ( u )where H is a unitary matrix ( H H = I ). Suppose e is an eigenvector of K X . a. Show He is an eigenvector of K Y . b. Show || He || = || e || . Problem 6. If Y ( u HX ( u )show R XY = R X H and R YX = HR X . Problem 7. Let W ( u )beawh iterandomvectorw ith μ W =(0 0 0) t , K W = I . Let X ( u HW ( u )+ c . Find c and a causal matrix H using the direct method that produces μ X = [ 123 ] T , K X = 111 122 .
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This note was uploaded on 09/13/2008 for the course EE 562a taught by Professor Toddbrun during the Summer '07 term at USC.

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EE562a_HW_1 - Homework 1 Due Monday June 5, 2006 Work the...

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