EL6303ClassNotes

EL6303ClassNotes - Probability EL6303- Section 2240 NYU-...

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Unformatted text preview: Probability EL6303- Section 2240 NYU- Poly Class Notes P. R. Moon October 15, 2010 2 Contents 0.1 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Basics 9 1.1 Classical Probability (Likelihood or Chance) . . . . . . . . . . . . . . 9 1.2 Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Axiomatic Probability Assignment . . . . . . . . . . . . . . . . . . . . . 16 1.4 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 APPENDIX: Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.2 Sets, Subsets, and Equality: Examples . . . . . . . . . . . . . . . . . 23 1.5.3 Basic Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5.4 Boolean Algebra and Truth Tables . . . . . . . . . . . . . . . . . . . 25 1.5.5 Properties of Set Operations . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.6 DeMorgan&s Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.7 Disjoint Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5.8 Decomposition of Space . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.5.9 Generalizations ( [ and \ ) . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5.10 Limits ( Q &! 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6 APPENDIX: Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.6.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.6.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.6.3 Case III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.6.4 Case IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.6.5 Occupancy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2 Conditional Probability, Statistical Independence, Total Probability, and Bayes Rule 41 2.1 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.3 Total Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 Bayes&Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 Compound Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3 4 CONTENTS 3 Random Variables 55 3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Repeated Trials (Binomial Random Variable) . . . . . . . . . . . . . . . . . 57 3.3 Geometric Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3....
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EL6303ClassNotes - Probability EL6303- Section 2240 NYU-...

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