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hw3 - Boston University Department of Electrical and...

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Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Problem Set No. 3 Fall 2008 Issued: Wednesday, Sept. 17, 2008 Due: Friday, Sept. 26, 2008 Problem 3.1 Let Z be a random variable with the following exponential density and distribution functions: p Z ( z ) = αe - αz , P Z ( z ) = 1 - e - αz where α > 0 is a parameter. Define the random process: X ( t ) = 1 if 0 t z 0 if t > z (a) Find the equivalent event for Z that corresponds to X ( t ) = 1, assuming a fixed t > 0. Use this to find the first-order marginal density p X ( t ) ( x ). (b) Assuming 0 < t 1 < t 2 , fill in the tables below with the equivalent events for Z and their corre- sponding probabilities to specify the second order marginal joint density function of X ( t 1 ) and X ( t 2 ), p X ( t 1 ) ,X ( t 2 ) ( x 1 , x 2 ). Equivalent Events p X ( t 1 ) ,X ( t 2 ) ( x 1 , x 2 ) X ( t 1 ) 0 1 0 X ( t 2 ) 1 X ( t 1 ) 0 1 0 X ( t 2 ) 1 Problem 3.2 (Old Exam Problem) Let α and β be two statistically independent, identically distributed Gaussian random variables with means E [ α ] = E [ β ] = 0 and variances σ 2 α = σ 2 β = 1. Define the stochastic process X ( t ) = α cos( t ) + β sin( t ). (a) Find the mean m X ( t ) and autocorrelation R XX ( t 1 , t 2 ) of the process X ( t ). (b) Is the process X ( t ) wide-sense stationary? Explain. (c) Is the process X ( t ) a Gaussian random process? Explain. Problem 3.3 (Old Exam Problem) Consider the random process X ( n ) = Z n even - Z n odd where Z N ( m, 1). Be sure your answers are valid for all possible values of m and to give explanations for your answers. (a) Find the mean and autocovariance functions of X ( n ). (b) Is X ( n ) wide-sense stationary? (c) Is X ( n ) a Markov process? (d) Is X ( n ) strict-sense stationary? 1
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(e) Is X ( n ) a Gaussian random process? (f) Is X ( n ) an independent increments process? Problem 3.4 Let X ( t ) = cos(2 πf 0 t + Φ) where f 0 > 0 is a constant and Φ is a random variable with p Φ ( φ ) = 1 4 δ ( φ ) + δ φ - π 2 + δ ( φ - π ) + δ φ - 3 π 2 (a) Find m X ( t ) and K XX ( t, s ). Is X ( t ) wide-sense stationary? (b) Find p X ( t ) ( x ). Is X ( t ) strict-sense stationary?
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