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ps7 - 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 7 Spring 2004 Issued: Thursday, April 1, 2004 Due: Thursday, April 8, 2004 Reading: For this problem set: Sections 4.3-4.6, 4.A, 4.B, Chapter 5 Next: Chapter 5, Sections 6.1 and 6.3 Problem 7.1 Let N ( t ) be a Poisson counting process on t 0 with rate . Let { y i } be a collection of statistically independent, identically-distributed random variables with mean and variance E [ y i ] = m y var y i = ω y 2 , respectively. Assume that the { y i } are statistically independent of the counting pro- cess N ( t ) and define a new random process y ( t ) on t 0 via 0 N ( t ) = 0 N ( t ) y ( t ) = . y i N ( t ) > 0 i =1 (a) Sketch a typical sample function of N ( t ) and the associated typical sample function of y ( t ). (b) Use iterated expectation (condition on N ( t ) in the inner average) to find E [ y ( t )] and E [ y 2 ( t )] for t 0. (c) Prove that y ( t ) is an independent-increments process on t 0 and use this fact to find the covariance function K yy ( t, s ) for t, s 0. Problem 7.2 (practice) (a) Let x ( t ) be an independent increments process on t 0 whose covariance func- tion is K xx ( t, s ), for t, s 0. Show that K xx ( t, s ) = var[ x (min( t, s ))] for t, s 0 . 1
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(b) Suppose x ( t ) in part (a) has stationary increments. Show that m x ( t ) = at + b, for t 0 , and K xx ( t, s ) = c min( t, s ) + d for t, x 0 where a, b are constants and c, d are non-negative constants.
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