# hw8_rp - Consider the random process W t = Xcos(2 πf t Y...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Vishwanath Homework 8 (Due 04/15/2009 in class) Problem1 The count of students dropping the course “Probability and Stochastic Processes” is known to be a Poisson process of rate 0 . 1 drops per day. Starting with day 0 , the ﬁrst day of the semester, let D ( t ) denote the number of students that have dropped after t days. What is p D ( t ) ( d ) ? Problem 2 For an arbitrary constant a , let Y ( t ) = X ( t + a ) . If X ( t ) is a stationary random process, is Y ( t ) stationary? Problem 3 Let X ( t ) be a stationary continuous time random process. By sampling X ( t ) every Δ seconds, we obtain the discrete time random sequence Y ( n ) = X ( n Δ) . Is Y ( n ) a stationary random sequence? Problem 4
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Unformatted text preview: Consider the random process W ( t ) = Xcos (2 πf t ) + Y sin (2 πf t ) where X and Y are uncorrelated random variables each with expected value and variance σ 2 . Find the autocorrelation R W ( t, τ ) . Is W ( t ) wide sense stationary? Problem 5 X ( t ) is a wide sense stationary random process with average power equal to 1 . Let Θ denote a random variable with uniform distribution over [0 , 2 π ] . such that X ( t ) and Θ are independent. a) What is E [ X 2 ( t )] ? b) What is E [ cos (2 πf c t + Θ)] ? c) Let Y ( t ) = X ( t ) cos (2 πf c t + Θ) . What is E [ Y ( t )] ? d) What is the average power of Y ( t ) ?...
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