Unformatted text preview: Discrete Structures II : Introduction
Reading for next meeting: Ross, Ch 1., Sec. 1. Rosen, Ch 5., Sec. 1 01/21/2009
Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 1 Today Class logistics What is Discrete Structures II about? Why you should know about Discrete Structures II? 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 2 Instructor & TAs Vladimir Pavlovic [email protected] Office: CoRE 312 Office hours: Thu. 3:00  5:00 Phone: 732 445 2001 x2654 http://www.cs.rutgers.edu/~vladimir TAs Section 1: Pavel Kuksa, [email protected], office hours TBA Section 2: Md Pavel Mahmud, [email protected], office hours TBA 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 3 Textbooks A First Course in Probability Sheldon Ross Prentice Hall, 7th ed., 2006. http://www.pearsonhighered.com/educator/academic/product/ 0,3110,0131856626,00.html Discrete Mathematics and its Applications Kenneth A. Rosen McGraw Hill, 6th ed., 2006 (Ch. 56), http://www.mhhe.com/math/advmath/rosenindex.mhtml 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 4 A First Course in Probability Sheldon Ross Prentice Hall, 7th ed., 2006. http:// cator/academic/product/0,3110,0131856626,00.html Discrete Mathematics and its Applications Kenneth A. Rosen McGraw Hill, 6 p://www.mhhe.com/math/advmath/rosenindex.mhtml Homework Homework will be assigned approx. every week to two weeks. You should work on the assignment independently. After all, homework is there and you learn and understand the topics we cover and prepare Policies to helpProcedures for the tests. ork At the time homework is assigned you will be given the due date. All homework is to be turned in by 11:59pm on the due date using cs206 be assigned approx. every week to two weeks. You should work on the assignme SAKAI site. ork is there to help you learn and understand the topics we(in PDF/PS). No MS the Format: PDF, PS, plain text, scanned handwritten cover and prepare for igned you will be given the due date. All homework is to be turned in by 11:59pm Word or other text processors. s accepted with the following penalty: the following penalty: Late homework is accepted with if you turned in your assignmentCS206  Intro. to Discreteday after the due date 5and you so at 2:10am the Structures II 01/21/2009 llWednesday, January 21, 2009 get a score of 80 instead of 100. See Grading Policy for further details. Hours Late (0,24] (24,48] (48, ) Penalty 20% 50% 100% if you turned in your assignment at 2:10am the day after the due date and you s get a score of 80 instead of 100. See Grading Policy for further details. Grading Criteria (48, ) 100% ccording to the table below. midterm and final exams. Each performance item will weigh according ill be based on how well you perform on homeworks, midterm and final exams. to the table below. You Criteria final grade will be based on how well you perform on homeworks, 30% 30% 30% 10% Homework Mindterm Final Quizzes scale n in an assignment (or a test) you will be given a numeric score S between 0 (wo n be converted to a normalized score: 3 S  E[S] Sn =  Intro. to Discrete Structures II + 4. 01/21/2009 CS206 6 2 StdDev[S]
Wednesday, January 21, 2009 Grading scale Grading Scale 30% 30% 30% 10% 30% 10% Homework Final Mindterm Quizzes Final Quizzes me you turn in an Grading scale assignment (or a test) you will be given a numeric score S between 0 (worst) an Each time you turn in an assignment (or a test) you will be given a e will then be converted to a normalized score: me you turn in an assignment (or a test) you(worst) given 100 (best). The score will thenand 1 numeric score S between 0 will be and a numeric score S between 0 (worst) 3 e will then be converted to aanormalized score: S  E[S] + 4. be converted to normalized score: Sn = StdDev[S] 3 2S  E[S] Sn = + 4. d to the range [0,7]. This, for instance, means that if you score 60 points on the final, the class 2 StdDev[S] standardrange [0,7]. to the range [0,7]. This, will you4.5. This formulathat beyou the class ave deviation is 30, your normalized score for be score 60 points can adjusted to d to the truncated This, for instance, means that if instance, means on theiffinal, score guara ance possible. standard 60 points on the final, the class average isThis and the standard deviation is 30, your normalized score will be 4.5. 50 formula can be adjusted to guarante normalized score will30, your normalized score will be 4.5. This formula canthe examp deviation is correspond to the letter grades in the table below. Hence, in be ance possible. orrespond toscore will guarantee to the letter grades in the table below. Hence, in the example adjusted to correspond the best performance possible. normalized letter grade B. orrespond to letter grade B. A 6.0  7.0 5.0  6.0 AB+ 6.0  7.0 B 4.0  5.0 B+ 5.0  6.0 3.0  4.0 BC+ 4.0  5.0 C 2.0  3.0 C+ 3.0  4.0 CD 2.0  3.0 1.0  2.0 DF 1.0  2.0 0.0  1.0 F 0.0  1.0 01/21/2009 CS206  Intro. to Discrete Structures II 7
Wednesday, January 21, 2009 2 Grading Scale (cont'd) Final course grade will be assigned based on the weighted average score of all assignments and tests:
F Sn = 0.3[Sn (HW 1) + ...Sn (HW N )]/N + 0.3Sn (M id) + 0.3Sn (F nl) + 0.1Sn (Quiz). l course grade will be assigned based on the weighted average score of all assignments and tests: r computing the final score I may adjust it based on my overall impression of your performance. etter grade will computing the final final numeric score F Sn and the tablemy overall After be computed from the score I may adjust it based on above. akai impression of your performance. Your final course letter grade will be computed from the final numeric score FSn and the table above. be using http://sakai.rutgers.edu online course management for posting announcemen signments, grades, and pretty much all communication during this semester. Familiarize yourself w ave not used it before. rst Week
01/21/2009 CS206  Intro. to Discrete Structures II 8 Wednesday, January 21, 2009 ill be no recitations the first week of classes, Jan. 19  Jan 23. Recitations Very important, you should attend! Practice problems, clarify confusing statements made by Professor (me). No recitations this week. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 9 What will you learn in 198:206? 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What will you learn in 198:206? How to... 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What will you learn in 198:206? How to... Count. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What will you learn in 198:206? How to... Count. Measure. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What will you learn in 198:206? How to... Count. Measure. Divide / Compute ratios. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What will you learn in 198:206? How to... Count. Measure. Divide / Compute ratios. Improve your betting skills. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 10 What (more) will you learn? 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). Conditional Probability, Bayes Theorem, Independence. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). Conditional Probability, Bayes Theorem, Independence. Random Variables. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). Conditional Probability, Bayes Theorem, Independence. Random Variables. Bernoulli Trials. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). Conditional Probability, Bayes Theorem, Independence. Random Variables. Bernoulli Trials. Expectation, Variance. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 What (more) will you learn? Combinatorics and Counting. Probability: Basic Ingredients (random experiment, sample space, events, probability measure). Conditional Probability, Bayes Theorem, Independence. Random Variables. Bernoulli Trials. Expectation, Variance. Applications of Probability and Combinatorics. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 11 So who cares? Case 1. Finding the odds of winning that poker game. Royal Flush? Straight flush? 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 12 So who cares? Case 2. An urn contains n balls, numbered 1, 2, . . . , n, which are drawn out one at a time. Find the probability that there are no matches in this process; that is, that ball i is not the ith ball drawn. The number of ways to arrange n couples around a table so that the sexes alternate and no husband and wife are seated together. Assigning n job applicants to m open positions, under constraints that include their qualifications, ... Rook Polynomials
01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 13 So who cares? Case 3. Suppose there are six people at a party, where each pair of people are either friends or strangers. How likely is it that there are going to be three mutual friends or three mutual strangers? What is the minimum number of people needed at a party in order to guarantee that there are at least four mutual friends or four mutual strangers? Ramsey Numbers. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 14 So who cares? Case 4. Suppose m = a + b votes were cast in an election, with candidate A receiving a votes and candidate B receiving b votes. The ballots are counted individually in some random order, giving rise to a sequence of a As and b Bs. What is the number of all such possible sequences? Assuming all such sequences are equally likely, what is the probability that candidate A led throughout the counting of the ballots? Catalan numbers. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 15 So who cares? Case 4. Consider the game of tictactoe. How many 204 possible game board configurations are there? Applications of Discrete Mathematics What if 1. Three equivalent some are the "same" (e.g., Figure we realize that tictactoe configurations. rotations of the board). How many by calling We can formalize this intuitive concept of "sameness"then? two tictactoe configurations congruent if one can be obtained from the other by Polya counting. a rotation or by a reflection. Congruence is an equivalence relation, in the
usual sense, and the equivalence classes into which it partitions the set of all configurations of the tictactoe board are called congruence classes. Thus, we seek the solution to the following problem:
CS206  Intro. to Discrete Structures II CongruenceClass Counting Problem (CCCP): Count the number of different Wednesday, January 21, 2009 congruence classes among the configurations with five 01/21/2009 16 So who (really) cares? Case 5, Classifying a set of objects according to some criteria: Accidents according to the day of the week on which they occurred; People according to their profession, age, sex, or nationality; Printing jobs according to the printer on which they were done. Function from the set of objects to the set of categories / levels. What is the number of different classifications possible? What is the probability that classifications satisfy certain conditions, e.g., that a job will be sent to (faulty) printer F? Given a set of objects how likely is it that two objects with the same category will be placed in the same set? Stirling numbers. 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 17 So who (finally) cares? Discover the structure of biological sequences, which can explain how cell processes are regulated, how likely cells are to be infected, etc. www.cs.cmu.edu/~epxing
01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 18 Topics & Schedule Introduction. Course organization. Requirements. Motivation. Discrete & continuous problems. Counting. Basic principles of counting. Inclusionexclusion. Sampling I. Sampling without replacement. Ordered sampling permutations. Unordered sampling combinations.
01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II 19 Sampling II. Ordered sampling with replacement. Binomial coefficient. Binomial theorem. Sampling III. Stirlings formula. Unordered sampling with replacement. Multinomial coefficient. Topics & Schedule (cont'd) Probability I. Subjective and objective theories of probability. Events. Sample spaces. Axioms of probability. Probability II. Properties of probability. Sample spaces with equally likely outcomes. Probability as a measure of belief. MIDTERM. Conditional Probability. Bayes' Rule. Bayes' formula. Evidence and updating of belief. Independence. Definition. Conditional independence. Random Variables. Discrete. Continuous. Distribution functions. Expectation. Functions of rv's. Expectation.
20 01/21/2009 Wednesday, January 21, 2009 CS206  Intro. to Discrete Structures II Topics & Schedule (cont'd) Variance. Properties. Other functions of rv's. Bernoulli Random Variable. Binomial Random Variable. Poisson Random Variable. Geometric Random Variable. Continuous Random Variables. Introduction. Expectation and variance. Density function. Geometric interpretation.
01/21/2009 Wednesday, January 21, 2009 Uniform Distribution. Gaussian Distribution. Joint Distribution.  Definition.  Examples.  Conditional distributions.  Independent rv's. Limit Theorems  Chebychev's inequality.  Weak law of large numbers.  Central limit theorem.  Strong law of large numbers. FINAL
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 Spring '08
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 Probability theory

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