hw3 - EE 562a Homework Set 3 Due Wednesday 14 Feb 2007...

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Unformatted text preview: EE 562a Homework Set 3 Due Wednesday 14 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 real random vector with mean vector m y and covariance matrix K y given by m y = 10 10 K y = 2- 1- 1 2 . (a) Sketch the regions of the y-plane to which each of the following events pertain, and numerically upper bound the probabilities of these events. A = { u ∈ U : y ( u, 1) < } B = { u ∈ U : [ y ( u, 1)- y ( u, 2)] / 2 > 20 } C = { u ∈ U : | y ( u )- m y | > 10 } (b) Given the additional information that y ( u ) is a Gaussian random vector, determine precise values of the probabilities of the events A and B above. (You may need a table of tail integrals of the one-dimensional Gaussian density to do this, e.g., Table 26.1 in M. Abramowitz and I. Stegun, Handbook of Mathematical Functions . If the table does not go far enough to suit your purpose, any of the asymptotic expansions 26.2.12-26.2.15 may give you an accurate answer with a small amount of work.) (c) Given the additional information that y ( u ) is a Gaussian random vector, determine upper and lower bounds to the probability of the event C above. There are several ways to develop such bounds. For example, you could inscribe and circumscribe appropriate ellipses around the region of integration C and apply the results of problem 1. Or youand apply the results of problem 1....
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hw3 - EE 562a Homework Set 3 Due Wednesday 14 Feb 2007...

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