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EE562a_Final2006_Solutions

# EE562a_Final2006_Solutions - EE 562a Final Solutions August...

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EE 562a Final Solutions August 7, 2006 Inst: Dr. C.W. Walker Problem Points Score 1 10 2 12 3 12 4 10 5 10 6 10 7 10 8 10 9 16 Total 100

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Problem 1. Suppose U is a random variable uniformly distributed in the interval [0, 1]. Let u be a realization of U . For each of the following state whether the sequence of random variables converges surely, almost surely or neither of these. If the sequence does converge surely or almost surely indicate the random variable or constant to which the sequence converges. a. X n ( u )= u n . Solution: almost surely to 0 b. Y n ( u u . Solution: surely to u c. V n ( u u n +( 1) n u n 2 . Solution: almost surely to 0 d. Z n ( u )=1+ u + u 2 + u 3 + ··· + u n . Solution: almost surely to 1 1 U Problem 2. We are given an observation of X and we must decide between two hypotheses: H 0 : X = N H 1 : X = S 1 + N where, N has density f N ( n 1 2 · e −| n | , −∞ <n< and S 1 =7 . 00. a. Find a threshold T so that the probability of type I error is 0.001. Solution: Z T f N ( n ) dn =10 3 T =6 . 21 b. Using the T you found ±nd the probability of type II error. Solution: Z T −∞ 1 2 e −| n T | dn = Z 6 . 21 −∞ 1 2 e ( n T ) dn =0 . 23 c. What is the power of this test? Solution: 1 0 . 23 = 0 . 77. 1
Problem 3. Consider a real Gaussian random sequence x ( n ), n an integer, with E [ x ( n )] = 0 ,E h x ( n ) 2 i =1 [ x ( n ) x ( m )] = ρ | n m | where 0 <ρ< 1. Let y ( n )= x ( n ) n +1 ,n 6 = 1 0 = 1 . a. Is x ( n ) wide sense stationary? Solution: yes. b. Find the covariance of y ( n ) and state whether or not y ( n )isw idesense stationary. Solution: E [ y ( n )] = 0 , K Y ( m, n E [ y ( m ) y ( n )] = ρ | m n | ( m +1)( n +1) ,m 6 = 1 6 = 1 0 ,e l s e . c. Does x ( n ) converge in the mean square sense and, if so, what is the limit?

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EE562a_Final2006_Solutions - EE 562a Final Solutions August...

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