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hw2 - EE 562a Homework Set 2 Due Monday 5 Feb 2007 1(The...

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EE 562a Homework Set 2 Due Monday 5 Feb 2007 1 (The following well-defined problems come from different sources, and the notation used may vary. Don’t let that bother you!) 1. Let y ( u ) be a random vector with second-moment description m y = 0 K y = 3 - 1 0 - 1 3 0 0 0 3 . (a) What is the mean squared length of y ( u )? (b) For what unit vector b max does y t ( u ) b max have maximum variance? (c) For what unit vector b min does y t ( u ) b min have minimum variance? (d) Upper bound the P {| y ( u ) | > 10 } . (Here P {·} denotes the probability of the event {·} .) 2. Find the biased linear transformations of a random vector w ( u ), composed of mean-zero, unit-variance, uncorrelated random variables, which will yield random vectors having the following second moment descriptions: (a) m x = 1 2 3 K x = 1 1 1 1 2 2 1 2 3 (b) m y = 1 1 - 1 - 1 K y = 1 - 1 0 0 - 1 1 0 0 0 0 2 - 2 0 0 - 2 2 (c) m z = 1 1 1 K z = 9 6 3 6 13 8 3 8 14 Hint: Parts (a) and (c) are most easily done with triangular-matrix transformations of w ( u ). For experience, part (b) ought to be done by the eigenvector method.
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