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TM605WS - Probability for Telecommunications Weeks/Lectures 5, 7 - 11 / Chapters 5, 6, 8 - 10 Prof. Thomas F. Brantle Final Exam - Part II Final Exam...

A consumer is trying to decide between two long-distance calling plans. The first charges a flat rate of 10¢ ($0.10) per minute, whereas the second charges a flat rate of 99¢ ($0.99) for calls up to 20 minutes in duration and then 10¢ ($0.10) for each additional minute exceeding 20 (assume that calls lasting a non-integer number of minutes are charged proportionately to a whole-minutes charge). Suppose the consumer’s distribution of call duration is exponential with parameter λ.
a. Explain intuitively how the choice of calling plan should depend on what the expected call duration is.
b. Which plan is better if expected call duration is 10 minutes? 15 minutes? [Hint: Let h1(x) denote the cost for the first plan when call duration is x minutes and let h2(x) be the cost function for the second plan. Give expressions for these two cost functions, and then determine the expected cost for each plan.]
TM605WS – Probability for Telecommunications Prof. Thomas F. Brantle Weeks/Lectures 5, 7 – 11 / Chapters 5, 6, 8 – 10 Final Exam – Part II 1 Final Exam Part II Weeks/Lectures 5, 7 11 / Chapters 5, 6, 8 10 This is an open book and open notes exam. You are expected to work independently and not share information. The point value for each question is three (3 ) points. The total number of points on this exam is 45 (includes 5 extra credit points). Please show all work and required explanation for full and/or partial credit as applicable. Your final submission should be a single comprehensive document in either Word or PDF format with any Excel, Jpeg, or other file types or scanned results copied/pasted into the submitted document. All work should be readable and legible to receive credit. Place your “Name” both within the document as well as within the file’s name. The due date and time for this exam is Sunday Midnight PT May 9, 2010 (3:00AM ET May 10, 2010) . Please let me know if you have any questions.
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TM605WS – Probability for Telecommunications Prof. Thomas F. Brantle Weeks/Lectures 5, 7 – 11 / Chapters 5, 6, 8 – 10 Final Exam – Part II 2 1. Suppose the reaction temperature X (in ) in certain chemical process has a uniform distribution with A = 5 and B = 5. a. Compute P ( X < 0). b. Compute P ( 2.5 < X < 2.5). c. Compute P ( −2 ≤ X ≤ 3). d. For k satisfying 5 < k < k + 4 < 5, compute P ( k < X < k +4). 2. The article “Modeling Sediment and Water Column Interactions for Hydrophobic Pollutants” ( Water Research , 1984: 1169 1174) suggests the uniform distribution on the interval (7.5, 20) as a model for depth (cm) of the bioturbation layer in sediment in a certain region. a. What are the mean and variance of depth? b. What is the CDF of depth? c. What is the probability that the observed depth is at most 10? Between 10 and 15? d. What is the probability that the observed depth is within 1 standard deviation of the mean value? Within 2 standard deviations? 3. Let X denote the distance (m) that an animal moves from its birth site to its first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, X has an exponential distribution with parameter λ = .01386 (as suggested in the article “Competition and Dispersal from Multiple Nests,” Ecology , 1997: 873 883. a. What is the probability that the distance is at most 100 m? At most 200 m? Between 100 and 200 m? b. What is the probability that distance exceeds the mean distance by more than 2 standard deviations? c. What is the value of the median distance? 4. A consumer is trying to decide between two long-distance calling plans. The first charges a flat rate of 10¢ ($0.10) per minute, whereas the second charges a flat rate of 99¢ ($0.99) for calls up to 20 minutes in duration and then 10¢ ($0.10) for each additional minute exceeding 20 (assume that calls lasting a non-integer number of minutes are charged proportionately to a whole- minutes charge). Suppose the consumer’s distribution of call duration is exponential with parameter λ . a. Explain intuitively how the choice of calling plan should depend on what the expected call duration is. b. Which plan is better if expected call duration is 10 minutes? 15 minutes? [Hint: Let h 1 (x) denote the cost for the first plan when call duration is x minutes and let h 2 (x) be the cost function for the second plan. Give expressions for these two cost functions, and then determine the expected cost for each plan.] 5. Suppose the number X of tornadoes observed in a particular region during a 1-year period has a Poisson distribution with λ = 8. a. Compute P ( X ≤ 5). b. Compute P (6 ≤ X ≤ 9). c. Compute P (10 ≤ X ).
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