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Unformatted text preview: 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 ) = e- z , P Z ( z ) = 1- e- z where > 0 is a parameter. Define the random process: X ( t ) = 1 if 0 t z 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 ) 1 X ( t 2 ) 1 X ( t 1 ) 1 X ( t 2 ) 1 Problem 3.2 (Old Exam Problem) Let and be two statistically independent, identically distributed Gaussian random variables with means E [ ] = E [ ] = 0 and variances 2 = 2 = 1. Define the stochastic process X ( t ) = cos( t ) + 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 (e) Is X ( n ) a Gaussian random process? (f) Is X ( n ) an independent increments process? Problem 3.4 Let X ( t ) = cos(2 f t + ) where f > 0 is a constant and is a random variable with p ( ) = 1 4 ( ) + - 2 + ( - ) + - 3 2 (a) Find m X ( t ) and K XX ( t, s ). Is X ( t ) wide-sense stationary?...
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