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Name (1%):Massachusetts Institute of TechnologyDepartment of Electrical Engineering and Computer ScienceDepartment of Mechanical Engineering6.050J/2.110J Information and Entropy Spring 2004Issued: May 21, 2004, 1:30 PM Final Exam Due: May 21, 2004, 4:30 PMNote: Please write your name at the top of each page in the space provided. The last page may be removedand used as a reference table for the calculation of logarithms.Problem 1: Information and Jeopardy (9%)From the list of names from Table 1 below, fill in the blank with the name that best fits each of the followingquestions. Make sure your answers to each part are different.a. This physicist of the 19th century has a unit of temperature named after him. Who was?b. This mathematician, whose name sounds suspiciously edible, invented a measure of how far twocodewords are from each other. Who was ?c. He was one of three after whom are named a compression technique used in the popular GIFimage format. Who was ?d. This Yale mathematician received the first doctorate in engineering awarded in America, and hasa famous inequality that b ears his name. Who was ?e. After reportedly secluding himself in his mountain cabin, with pearls in his ears to muffle thesound, and his girlfriend in his bed to inspire him, this famous physicist came up with an equationwhich bears his name that is used to calculate the wavefunction of a quantum system. Who was?f. The constant named after this physicist, who committed suicide in 1906 purportedly because hisscientific ideas were not accepted by his peers, has a value of approximately 1.38 × 10−23Joulesper Kelvin. Who was ?g. This military engineer showed, through the quantity that bears his name, that the efficiency ofa heat engine can never be 1. Who was ?h. Known most famously for the form of algebra named after him, this mathematician was a child-hood prodigy in Latin, publishing at the age of 12. Who was ?i. While at MIT as a graduate student in the 1940’s this man invented a famous inequality whichbears his name. Who was ?Avogadro Bayes Boltzmann Boole Carnot Gibbs HammingHuffman Jaynes Joule Kelvin Kraft Lempel MaxwellMorse Reed Schr¨odinger Shannon Solomon Welsh ZivTable 1: List of NamesFinal Exam; Name: 2Problem 2: MIT Customer Complaint Department (15%)You have recently been elected President of the UA, and it is your job to transmit student complaints toChuck Vest so they (hopefully) can be addressed before he steps down. According to the UA’s research, allstudent complaints fall into one of six categories, with percentages shown:% Complaint50% Not enough homework30% Campus dining options too diverse10% Tuition to o low5% Administration too attentive5% Classes too easyUnfortunately, Vest doesn’t have much time, so he instructs you to only send very short messages to him.Because you’ve taken 6.050 you know about coding schemes, so you decide to encode the complaints abovein a Huffman code.a. Design a Huffman co de for the complaints above.Complaint CodeNot enough homeworkCampus dining options too diverseTuition to o lowAdministration too attentiveClasses too easyb. What is the average number of bits to send one complaint?Average # of bits/complaint:Final Exam; Name: 3Problem 3: The Traveling SailMan (20%)Nothing annoys owners of expensive sailboats more than dirty sails. The SailMan robot solves this problem.Every night it swims from boat to boat, climbs aboard, and cleans the sails. This robot is helped byreconnaissance robots that report the location of boats with dirty sails to a central computer, which thenplans the night’s activities by solving the “Traveling SailMan” problem. Because of software limitations,reconnaissance robots can only be deployed in groups of 2 (Small), 4 (Medium), or 8 (Large).Visiting a marina one evening, you noticed the SailMan robot and naturally wondered how many recon-naissance robots were deployed there. You expressed your knowledge in terms of probabilities (call themS, M, and L) that there were 2, 4, or 8 reconnaissance robots. You remembered from the SailMan AnnualReport that the average number of reconnaissance robots deployed in a marina was 6.a. With this knowledge, what values of S, M, and L are possible?SminSmaxMminMmaxLminLmaxb. You decided to use the Principle of Maximum Entropy to estimate S, M , L, and the resulting un-certainty U about the number deployed. Express the entropy as a function of a single probability(any one of the three, S, M, or L).Entropy =Before you had a chance to evaluate U, you happ e ned to run into the the Fiscal Officer of SailMan, Inc., whosaid that when he wrote the Annual Report he wanted readers to think the high-priced configuration of 8reconnaissance robots was selling better than it really was. He reported the average number of reconnaissancerobots (6) correctly but then mentioned the largest value of L that was consistent with this average, withoutactually saying it was correct (linguists would call this an “intended inference”).c. What probabilities S, M, and L did he want readers to infer, and what would be their resultinguncertainty in bits about the number of reconnaissance robots deployed in any one marina?S = M = L = Uncertainty =d. Compare this uncertainty to the value of U which you started to find earlier in this problem. Isthis uncertaintyLess than U? Equal to U? Greater than U?Final Exam; Name: 4Problem 4: Variations on a Theme by Carnot (25%)The ideal (and most efficient) version of a magnetic heat engine operates reversibly in a cycle that can berepresented as a rectangle in the T − S plane. The same physical device can also be operated reversiblyas a refrigerator or as a heat pump to accomplish different goals. This problem asks you to quantify theeffectiveness of these three systems in terms of the variables in the diagram below. The entropies andtemperatures in the plot and the energies given in the description of each problem are all positive.Figure 1: T − S Cycle for a reversible heat engineHeat Engine: The purpose of a heat engine is to convert heat to work. In each cycle a heat enginereversibly extracts QaJoules of heat from a hot reservoir at temperature Th, dumps a portion QbJoules ofthat heat into a cooler environment at temperature Tcand converts the remaining energy into (Qa− Qb)Joules of useful work. The cost per cycle is the energy Qataken from the hot reservoir and the benefit percycle is the work (Qa− Qb) p e rformed.a. Does a heat engine traverse the


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