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Unformatted text preview: Boston University Department of Electrical and Computer Engineering EC505 STOCHASTIC PROCESSES Problem Set No. 4 Fall 2008 Issued: Wednesday, Sept. 24, 2008 Due: Friday, Oct. 3, 2008 Problem 4.1 (a) Let X 1 ( t ) be a random telegraph wave. Specifically, let N ( t ) be a Poisson counting process with Pr [ N ( t ) = k ] = ( t ) k e t k ! . Let X 1 (0) = +1 with probability 1/2 and X 1 (0) = 1 with probability 1/2, assume that X 1 (0) is independent of N ( t ) and define X 1 ( t ) = X 1 (0) if N ( t ) is even X 1 (0) if N ( t ) is odd Sketch a typical sample function of X 1 ( t ). Find m X 1 ( t ), K X 1 X 1 ( t, s ), p X 1 ( t ) ( x ), and p X 1 ( t )  X 1 ( s ) ( x t  x s ). You may find the following sums useful: X k =0 k k ! = e , X k = 0 k even k k ! = cosh( ) , X k = 0 k odd k k ! = sinh( ) . (b) Let X 2 ( t ) be a Gaussian random process with m X 2 ( t ) = 0 , and K X 2 X 2 ( t, s ) = e 2  t s  Find p X 2 ( t ) ( x ), and p X 2 ( t )  X 2 ( s ) ( x t  x s ). Sketch a typical sample function of x 2 ( t ). Show that X 2 ( t ) is not an independent increments process. Note: The processes in (a) and (b) have the same second order properties, yet have very different sample paths, first order densities, etc. Problem 4.2 Let N ( t ) be a Poisson counting process on t 0 with rate . Let { Y i } be a collection of statistically independent, identicallydistributed random variables with mean and variance E [ Y i ] = m y and var[ Y i ] = 2 y respectively. Assume that the { Y i } are statistically independent of the counting process N ( t ) and define a new random process Y ( t ) on t 0 via y ( t ) = N ( t ) = 0 N ( t ) X i =1 Y i N ( t ) > (a) Sketch a typical sample function on N ( t ) and the associated typical sample function of Y ( t ). (b) Use smoothing property of expectations (condition on N ( t ) in the inner average) to find E [ Y ( t )] and E [ Y 2 ( t )] for t 0....
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 Fall '04
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