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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 ChampaignUrbana 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 ChampaignUrbana and Springfield. 40 Springfield ChampaignUrbana 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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 Spring '08
 Sarwate
 Variance, Probability theory, probability density function

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