fall09HEIISoln - University of Illinois Fall 2009 ECE 313...

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Unformatted text preview: University of Illinois Fall 2009 ECE 313: Hour Exam II Monday November 16, 2009 7:00 p.m. — 8:00 p.m. 100 Materials Science and Engineering Building 1. [15 points] Messages can be transmitted between Champaign-Urbana and Springfield either directly or via Decatur on communication links whose capacities are indicated on the diagram shown below. Assume that the three links fail independently with equal probability 1 2 . Let X denote the communication capacity between Champaign-Urbana and Springfield. 40 Springfield Champaign-Urbana Decatur 200 100 (a) [5 points] What values can X take on? Solution: It is easy to get that X can take on values 0, 40, 200, and 240. (b) [10 points] Find the expected value of X . Solution: X has value 240 when all links are working, which occurs with probability 1 8 . X has value 40 when the direct link between CU and Springfield has failed while the other two links are working, which also occurs with probability 1 8 . X has value 200 when the direct link between CU and Springfield is working while at least one of the other two links has failed, which occurs with probability 3 8 . Hence, E [ X ] = 240 × 1 8 + 200 × 3 8 + 40 × 1 8 = 240 + 3 × 200 + 40 8 = 880 8 = 110 . 2. [10 points] Let X denote a Gaussian random variable with mean 2 and variance 25. Use the table of values of the unit Gaussian CDF Φ( · ) on the last page of this exam booklet to find the numerical value of P {| X- 4 | > 3 } ....
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fall09HEIISoln - University of Illinois Fall 2009 ECE 313...

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