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# Probabilidades_Discretas - Elementary Discrete Probability...

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Elementary Discrete Probability David A. SANTOS [email protected] December 9, 2005 REVISION

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ii Contents Preface iii To the Student v 1 Preliminaries 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . 1 Homework . . . . . . . . . . . . . . . . . . . . . 2 1.2 Sample Spaces and Events . . . . . . . . . . 3 Homework . . . . . . . . . . . . . . . . . . . . . 4 1.3 Combining Events . . . . . . . . . . . . . . . 4 Homework . . . . . . . . . . . . . . . . . . . . . 5 1.4 Functions . . . . . . . . . . . . . . . . . . . 6 Homework . . . . . . . . . . . . . . . . . . . . . 8 Answers . . . . . . . . . . . . . . . . . . . . . . 8 2 Counting 10 2.1 Inclusion-Exclusion . . . . . . . . . . . . . . 10 Homework . . . . . . . . . . . . . . . . . . . . . 13 2.2 The Product Rule . . . . . . . . . . . . . . . 14 Homework . . . . . . . . . . . . . . . . . . . . . 18 2.3 The Sum Rule . . . . . . . . . . . . . . . . . 19 Homework . . . . . . . . . . . . . . . . . . . . . 21 2.4 Permutations without Repetitions . . . . . . . 22 Homework . . . . . . . . . . . . . . . . . . . . . 23 2.5 Permutations with Repetitions . . . . . . . . 24 Homework . . . . . . . . . . . . . . . . . . . . . 26 2.6 Combinations without Repetitions . . . . . . 27 Homework . . . . . . . . . . . . . . . . . . . . . 30 2.7 Combinations with Repetitions . . . . . . . . 33 Homework . . . . . . . . . . . . . . . . . . . . . 35 2.8 Binomial Theorem . . . . . . . . . . . . . . 35 Homework . . . . . . . . . . . . . . . . . . . . . 38 2.9 Miscellaneous Counting Problems . . . . . . . 38 Homework . . . . . . . . . . . . . . . . . . . . . 40 Answers . . . . . . . . . . . . . . . . . . . . . . 41 3 Discrete Probability 47 3.1 Probability Spaces . . . . . . . . . . . . . . 47 Homework . . . . . . . . . . . . . . . . . . . . . 49 3.2 Uniform Random Variables . . . . . . . . . . 50 Homework . . . . . . . . . . . . . . . . . . . . . 55 3.3 Independence . . . . . . . . . . . . . . . . . 58 Homework . . . . . . . . . . . . . . . . . . . . . 60 3.4 Binomial Random Variables . . . . . . . . . . 61 Homework . . . . . . . . . . . . . . . . . . . . . 63 3.5 Geometric Random Variables . . . . . . . . . 63 Homework . . . . . . . . . . . . . . . . . . . . . 65 3.6 Poisson Random Variables . . . . . . . . . . 65 Answers . . . . . . . . . . . . . . . . . . . . . . 66 4 Conditional Probability 70 4.1 Conditional Probability . . . . . . . . . . . . 70 Homework . . . . . . . . . . . . . . . . . . . . . 71 4.2 Conditioning . . . . . . . . . . . . . . . . . 71 Homework . . . . . . . . . . . . . . . . . . . . . 75 4.3 Bayes’ Rule . . . . . . . . . . . . . . . . . . 76 Homework . . . . . . . . . . . . . . . . . . . . . 78 Answers . . . . . . . . . . . . . . . . . . . . . . 79 5 Expectation and Variance 81 5.1 Expectation and Variance . . . . . . . . . . . 81 Homework . . . . . . . . . . . . . . . . . . . . . 84 5.2 Indicator Random Variables . . . . . . . . . . 85 Homework . . . . . . . . . . . . . . . . . . . . . 86 5.3 Conditional Expectation . . . . . . . . . . . 86 Homework . . . . . . . . . . . . . . . . . . . . . 87 Answers . . . . . . . . . . . . . . . . . . . . . . 87 6 Markov Chains 88 6.1 Discrete Time Stochastic Processes . . . . . . 88 Homework . . . . . . . . . . . . . . . . . . . . . 89 6.2 Long Run Probabilities . . . . . . . . . . . . 89 Answers . . . . . . . . . . . . . . . . . . . . . . 90 7 Uniform Continuous Random Variables 91 A The Integers 96 Homework . . . . . . . . . . . . . . . . . . . . . 99 B Divisibility Tests 100 Homework . . . . . . . . . . . . . . . . . . . . . 103 C Arithmetic Sums 104 Homework . . . . . . . . . . . . . . . . . . . . . 108 D Geometric Sums 109 Homework . . . . . . . . . . . . . . . . . . . . . 112 Answers . . . . . . . . . . . . . . . . . . . . . . 112
Preface These notes started during the Spring of 2002. The contents are mostly discrete probability, suitable for students who have mastered only elementary algebra. No calculus is needed, except perhaps in a very few optional exercises. Since a great number of the audience of this course comprises future elementary school teachers, I have included a great deal of preliminary ancillary material, especially in the areas of arithmetic and geometric sums and divisibility criteria. It has been my experience that many of these future teachers do actually enjoy learning the fundamentals of number theory and divisibility through probability problems. The response overall, has been positive. I would appreciate any comments, suggestions, corrections, etc., which can be addressed at the email below. David A. Santos [email protected] Things to do: Weave functions into counting, à la twelfold way. . . Write a chapter on expectation and include conditional expectation. Write a chapter on Markov Chains. Write a chapter on Games. Make use of indicator random variables. Write a section on the Pascal distribution. iii

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Legal Notice This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, version 1.0 or later (the latest version is presently available at http://www.opencontent.org/openpub/ THIS WORK IS LICENSED AND PROVIDED “AS IS” WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE OR A WARRANTY OF NON-INFRINGEMENT.
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