Probabilidades_Discretas

Probabilidades_Discretas - Elementary Discrete Probability...

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Unformatted text preview: Elementary Discrete Probability David A. SANTOS dsantos@ccp.edu December 9, 2005 REVISION 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...
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Probabilidades_Discretas - Elementary Discrete Probability...

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