ps05 - University of Illinois Spring 2010 ECE 313 Problem...

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Unformatted text preview: University of Illinois Spring 2010 ECE 313: Problem Set 5 Conditional Probability, Law of Total Probability, Bayes’ Formula Due: Wednesday February 24 at 4 p.m. Reading: Ross Chapter 3; Powerpoint Lecture Slides, Sets 9-14 Noncredit Exercises: Chapter 3: Problems 1, 2, 5, 10, 12, 16, 31, 38, 39, 51, 52 Theoretical Exercises 1, 2, 8, 16; Self-Test Problems 1-14. Reminders: Exam February 25, No class on Friday February 26 but office hours as usual next week 1. [Definition of Conditional Probability] 60% of all apples in the supermarket are red. 30% of the produce in the supermarket is red, and 50% of the red produce items are apples. What fraction of all produce items in the supermarket are apples? 2. [Conditional Probability and Total Probability] Simplify the following expresions. (a) P ( A ∪ B ∪ C | BC ) (b) P ( BC | B C ∪ C C ) (c) P ( ABCD | E ) + P ( A C BCD | E ) (d) P ( A | BC ) P ( B | C ) P ( C ) + P ( A | B C C ) P ( B C | C ) P ( C ) 3. [Bernoulli Bus Lines] Once per minute, a bus arrives at Sixth and Green with probability P . Consecutive minutes are independent; knowing that a bus arrived at 4:37 tells you nothing about what will happen at 4:38. Buses arriving after 5:00PM always arrive in the following order: the Turquoise line (#16) always comes first, followed by the Fuchsia line (#102), followed by the Octarine (#42). Your bus is the Octarine. (a) You arrive at the bust stop at exactly 5:00. i. What is the probability that you will wait exactly k minutes for your bus? ii. What is your expected waiting time? iii. What is the variance of your waiting time? (b) Your 4:00 lecturer (being half elvish) has a poor grasp of time, therefore you reach the bus stop at 5:05PM. What is the probability that your bus arrives m minutes after you? Note that m may be negative. (c) Fortunately, your friend Joe has been at the bus stop since 5:00. He says you haven’t missed your bus; the Turquoise bus came at 5:03, but the Fuchsia and Octarine buses have not arrived yet. Assuming that Joe is reliable, what is the probability that you will wait m minutes for your bus?...
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ps05 - University of Illinois Spring 2010 ECE 313 Problem...

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