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# final06sol - 1 National Chiao Tung University Department of...

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1 National Chiao Tung University Department of Electronics Engineering Stochastic Processes, Final Exam Solutions NOTE: Remember to write down your name. There are 5 problem sets with 100 points in total. 1. True or False (5+5+5+5=20 points) Indicate whether the statement in each of the following is True or False. Please provide an explanation, either a concise proof or a counter example, to your answer. Otherwise, your answer will carry no credits. (a) True. We can prove this using Chebyshev’s inequality (p. 380 in textbook). (b) True. By the definition of independent increments, for all t 1 , t 2 , . . . , t N such that t 1 < t 2 < . . . < t N , we have X ( t 1 ) , X ( t 2 ) - X ( t 1 ) , . . . , X ( t N ) - X ( t N - 1 ) are mutually independent for every integer N > 1. If we choose N = 3, and the time points t 1 < s < t 2 , it is clear that X ( t 2 ) - X ( s ), X ( s ) - X ( t 1 ) and X ( t 1 ) are mutually independent. (c) True. Since X ( t ) has E [ | X ( t + T ) - X ( t ) | 2 ] = 0 for all t , we know R XX (0) = R XX ( T ). The Cauchy-Schwarz inequality tells that E h‡ X ( t + T + τ ) - X ( t + τ ) · X ( τ ) 2 E X ( t + T + τ ) - X ( t + τ ) · 2 E h X 2 ( τ ) i . The right-hand side of the above is zero. And the left-hand side is ( R XX ( t + T ) - R XX ( t )) 2 , which is greater than zero due to the square. It follows the left-hand side is zero, yielding R XX ( t + T ) = R XX ( t ) . (d) True. The autocorrelation function is R XX ( τ ) = E [ x ( t + τ ) x ( t )] = E [ x ( t ) x ( t + τ )] = R XX ( - τ ) . So autocorrelation function is even, which can be used to show that S XX ( ω ) is also even.

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2 2. (5+5+5+5=20 points) Answer the following questions. Please be concise and precise. (a) See page 512 for the definition of ergodic in the mean, or page 514 Theo- reme 8.4-1 in textbook.
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final06sol - 1 National Chiao Tung University Department of...

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