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Finan Department of Mathematics Arkansas Tech University  \n􀀍cAll Rights Reserved Revised 2020 Edition  \nIn memory of my parents  \nAugust 1, 2008 January 7, 2009  \nPreface  \nThe present manuscript is a revised edition of the text that appeared 􀀌rstin the year 2007 . It adheres to the the SOA Syllabus October June 2020 . The book is designed mainly to help students prepare for the Probability Exam (known as Exam P/1), the 􀀌rst 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. However, the text will include the mathematical background needed throughout the book.  \nUsers are encouraged to access the online site of the Society of Actuaries [www.soa.org](www.soa.org for)[ for](www.soa.org for) up-to-date information about the exam.  \nThe present version includes in many cases in depth discussions in terms of proofs which the reader can omit. For this reason, the book is suitable for a one or two-semester course in undergraduate probability theory.  \nUsers of the book who are preparing for the exam are strongly encouraged to work out all the problems in the book. Make sure you can come out with the solutions on your own without any outside reference. This will enhance your chances in performing well on the exam.  \nThe book covers all the sample problems from previous exams made available by SOA. Such problems will be indicated by the symbol z:  \nAnswer keys to text problems are found at the end of the book. Mock exams are included as well. Attempt these exams after 􀀌nishing all the chapters of the book. Take these exams under the same circumstances as the actual exam.  \nThe present version of the book is made possible by a grant support from Arkansas Tech University.  \nii PREFACE  \nFinally, this manuscript can be used for personal use or class use, but not for commercial purposes. If you 􀀌nd any errors, I would appreciate hearing from you: m􀀌nan@atu.edu  \nAll the best.  \nMarcel B. Finan Russellville, AR August, 2020  \nContents  \nPreface i  \n1 Set Theory Prerequisite 5  \n1.1 Some Basic De􀀌nitions ...................... 6  \n1.2 Set Operations .......................... 16  \n2 Counting and Combinatorics 31  \n2.1 The Fundamental Principle of Counting ............ 32  \n2.2 Permutations ........................... 39  \n2.3 Combinations ........................... 45  \n3 Review of Calculus 53  \n3.1 Limits and Continuity ...................... 54  \n3.2 Series ................................ 66  \n3.3 The Derivative of a Function ................... 73  \n3.4 The De􀀌nite Integral ....................... 83  \n3.5 Improper Integrals ........................ 95  \n3.6 Graphing Systems of Inequalities in Two Variables ...... 106  \n3.7 Iterated Double Integrals ..................... 112  \n4 Probability: De􀀌nitions and Properties 121  \n4.1 Basic De􀀌nitions and Axioms of Probability .......... 122  \n4.2 Probability of Intersection and Union .............. 132  \n4.3 Probability and Counting Techniques .............. 140  \n5 Conditional Probability and Independence 147  \n5.1 Conditional Probabilities ..................... 148  \n5.2 Bayes' Formula and the Law of Total Probability ....... 156  \n2 CONTENTS  \n5.3 Independent Events ........................ 166  \n5.4 Odds Versus Probability ..................... 176  \n6 Discrete Random Variables 181  \n6.1 Random Variables ......................... 182  \n6.2 Probability Mass Function and Cumulative Distribution Function ................................ 189  \n6.3 Expected Value of a Discrete Random Variable ........ 199  \n6.4 Expected Value of a Function of a Discrete Random Variable . 208  \n6.5 Variance and Standard Deviation of a Discrete Random Variable 217  \n7 Commonly Used Discrete Random Variables 225  \n7.1 Uniform","cbCaiiMBy4klTj7t","https://ap.wps.com/l/cbCaiiMBy4klTj7t","pdf",3829727,794,"English","# Preface\n# 1 Set Theory Prerequisite\n## 1.1 Some Basic Definitions\n## 1.2 Set Operations\n# 2 Counting and Combinatorics\n## 2.1 The Fundamental Principle of Counting\n## 2.2 Permutations\n## 2.3 Combinations\n# 3 Review of Calculus\n## 3.1 Limits and Continuity\n## 3.2 Series\n## 3.3 The Derivative of a Function\n## 3.4 The Definite Integral\n## 3.5 Improper Integrals\n## 3.6 Graphing Systems of Inequalities in Two Variables\n## 3.7 Iterated Double Integrals\n# 4 Probability: Definitions and Properties\n## 4.1 Basic Definitions and Axioms of Probability\n## 4.2 Probability of Intersection and Union\n## 4.3 Probability and Counting Techniques\n# 5 Conditional Probability and Independence\n## 5.1 Conditional Probabilities\n## 5.2 Bayes' Formula and the Law of Total Probability\n## 5.3 Independent Events\n## 5.4 Odds Versus Probability\n# 6 Discrete Random Variables\n## 6.1 Random Variables\n## 6.2 Probability Mass Function and Cumulative Distribution Function\n## 6.3 Expected Value of a Discrete Random Variable\n## 6.4 Expected Value of a Function of a Discrete Random Variable\n## 6.5 Variance and Standard Deviation of a Discrete Random Variable\n# 7 Commonly Used Discrete Random Variables\n## 7.1 Uniform Discrete Random Variable\n## 7.2 Binomial Random Variable\n## 7.3 The Expected Value and Variance of the Binomial Distribution\n## 7.4 Poisson Random Variable\n## 7.5 Poisson Approximation to the Binomial Distribution\n## 7.6 Geometric Random Variable\n## 7.7 Negative Binomial Random Variable\n## 7.8 Hyper-geometric Random Variable\n# 8 Cumulative and Survival Distribution Functions\n## 8.1 The Cumulative Distribution Function\n## 8.2 The Survival Distribution Function\n# 9 Continuous Random Variables\n## 9.1 Distribution Functions\n## 9.2 The Expected Value of a Continuous Random Variable\n## 9.3 The Variance of a Continuous Random Variable\n## 9.4 Median, Mode, and Percentiles\n## 9.5 The Continuous Uniform Distribution Function\n## 9.6 Normal Random Variables\n## 9.7 The Normal Approximation to the Binomial Distribution\n## 9.8 Exponential Random Variables\n## 9.9 Gamma Distribution\n## 9.10 The Distribution of a Function of a Continuous Random Variable\n# 10 Joint Distributions\n## 10.1 Discrete Jointly Distributed Random Variables\n## 10.2 Jointly Continuous Distributed Random Variables\n## 10.3 Independent Random Variables\n## 10.4 Order Statistics\n## 10.5 Sum of Two Independent Random Variables: Discrete Case\n## 10.6 Sum of Two Independent Random Variables: Continuous Case\n## 10.7 Conditional Distributions: Discrete Case\n## 10.8 Conditional Distributions: Continuous Case\n## 10.9 Joint Probability Distributions of Functions of Random Variables\n# 11 Properties of Expectation","[{\"question\":\"What exam does this Probability course prepare for?\",\"answer\":\"It is designed to help students prepare for the Society of Actuaries Probability Exam, known as Exam P/1.\"},{\"question\":\"What knowledge is assumed before studying the material?\",\"answer\":\"A thorough command of calculus is assumed, though the book also provides the mathematical background needed throughout.\"},{\"question\":\"How does the book support practice beyond standard lessons?\",\"answer\":\"It covers sample problems from previous SOA exams (marked with a symbol) and includes answer keys and mock exams to be attempted after finishing the chapters.\"}]","Lecture Notes in Actuarial Mathematics - A Probability Course for the Actuaries - A Preparation for Exam P/1 | PDF",1790493253,2001]