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Unformatted text preview: A Probability Course for the Actuaries A Preparation for Exam P/1 Marcel B. Finan Arkansas Tech University c All Rights Reserved Preliminary Draft Last updated January 26, 2011 2 In memory of my parents August 1, 2008 January 7, 2009 Preface The present manuscript is designed mainly to help students prepare for the Probability Exam (Exam P/1), the ﬁrst actuarial examination administered by the Society of Actuaries. This examination tests a student’s knowledge of the fundamental probability tools for quantitatively assessing risk. A thorough command of calculus is assumed. More information about the exam can be found on the webpage of the Society of Actauries www.soa.org . Problems taken from samples of the Exam P/1 provided by the Casual Society of Actuaries will be indicated by the symbol ‡ . The ﬂow of topics in the book follows very closely that of Ross’s A First Course in Probabiltiy . This manuscript can be used for personal use or class use, but not for commercial purposes. If you ﬁnd any errors, I would appreciate hearing from you: mﬁnan@atu.edu This manuscript is also suitable for a one semester course in an undergraduate course in probability theory. Answer keys to text problems are found at the end of the book. This project has been partially supported by a research grant from Arkansas Tech University. Marcel B. Finan Russellville, AR May 2007 3 4 PREFACE Contents Preface 3 Basic Operations on Sets 9 1 Basic Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Counting and Combinatorics 31 3 The Fundamental Principle of Counting . . . . . . . . . . . . . . 31 4 Permutations and Combinations . . . . . . . . . . . . . . . . . . . 37 5 Permutations and Combinations with Indistinguishable Objects . 47 Probability: Deﬁnitions and Properties 57 6 Basic Deﬁnitions and Axioms of Probability . . . . . . . . . . . . 57 7 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . 65 8 Probability and Counting Techniques . . . . . . . . . . . . . . . . 74 Conditional Probability and Independence 81 9 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . 81 10 Posterior Probabilities: Bayes’ Formula . . . . . . . . . . . . . . 89 11 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . 100 12 Odds and Conditional Probability . . . . . . . . . . . . . . . . . 109 Discrete Random Variables 113 13 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 113 14 Probability Mass Function and Cumulative Distribution Function119 15 Expected Value of a Discrete Random Variable . . . . . . . . . . 127 16 Expected Value of a Function of a Discrete Random Variable . . 135 17 Variance and Standard Deviation . . . . . . . . . . . . . . . . . 142 5 6 CONTENTS 18 Binomial and Multinomial Random Variables . . . . . . . . . . . 148 19 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 12/14/2011 for the course ECE 604 taught by Professor Kumar during the Spring '11 term at Indian Institute of Technology, Kharagpur.
 Spring '11
 KUMAR
 The Land

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