Lecture Notes in Actuarial Mathematics A Probability ...

1 Lecture Notes in ActuarialMathematicsA Probability Course for theActuariesA Preparation for Exam P/1 Marcel B. FinanDepartment of MathematicsArkansas Tech Universityc All Rights ReservedRevised 2020 EditionIn memory of my parentsAugust1, 2008 January 7, 2009 PrefaceThe present manuscript is a revised edition of the text that appeared firstin the year 2007. It adheres to the the SOA Syllabus October June book is designed mainly to help students prepare for the ProbabilityExam (known as Exam P/1), the first Actuarial examination administeredby the Society of Actuaries. This examination tests a student s knowledgeof the fundamental Probability tools for quantitatively assessing risk. Athorough command of calculus is assumed.

2 However, the text will includethe mathematical background needed throughout the are encouraged to access the online site of the Society of up-to-date information about the present version includes in many cases in depth discussions in terms ofproofs which the reader can omit. For this reason, the book is suitable for aone or two- semester course in undergraduate Probability of the book who are preparing for the exam are strongly encouragedto work out all the problems in the book. Make sure you can come out withthe solutions on your own without any outside reference. This will enhanceyour chances in performing well on the book covers all the sample problems from previous exams made availableby SOA.

3 Such problems will be indicated by the symbol .Answer keys to text problems are found at the end of the book. Mock examsare included as well. Attempt these exams after finishing all the chaptersof the book. Take these exams under the same circumstances as the present version of the book is made possible by a grant support fromArkansas Tech , this manuscript can be used for personal use or class use, but notfor commercial purposes. If you find any errors, I would appreciate hearingfrom you: the B. FinanRussellville, ARAugust, 2020 ContentsPrefacei1 Set Theory Some Basic Definitions .. Set Operations .. 162 Counting and The Fundamental Principle of Counting .. Permutations.

4 Combinations .. 453 Review of Limits and Continuity .. Series .. The Derivative of a Function .. The Definite Integral .. Improper Integrals .. Graphing Systems of Inequalities in Two Variables .. Iterated Double Integrals .. 1124 Probability : Definitions and Basic Definitions and Axioms of Probability .. Probability of Intersection and Union .. Probability and Counting Techniques .. 1405 Conditional Probability and Conditional Probabilities .. Bayes Formula and the Law of Total Probability .. Independent Events .. Odds Versus Probability .. 1766 Discrete random random Variables .. Probability Mass Function and Cumulative Distribution Func-tion.

5 Expected Value of a Discrete random variable .. Expected Value of a Function of a Discrete random variable . Variance and Standard Deviation of a Discrete random Variable2177 Commonly Used Discrete random Uniform Discrete random variable .. Binomial random variable .. The Expected Value and Variance of the Binomial Distribution Poisson random variable .. Poisson Approximation to the Binomial Distribution .. Geometric random variable .. Negative Binomial random variable .. Hyper-geometric random variable .. 2788 Cumulative and Survival Distribution The Cumulative Distribution Function .. The Survival Distribution Function .. 3039 continuous random Distribution Functions.

6 The Expected Value of a continuous random variable .. The Variance of a continuous random variable .. Median, Mode, and Percentiles .. The continuous Uniform Distribution Function .. Normal random Variables .. The Normal Approximation to the Binomial Distribution .. Exponential random Variables .. Gamma Distribution .. The Distribution of a Function of a continuous random Variable389 CONTENTS310 Joint Discrete Jointly Distributed random Variables .. Jointly continuous Distributed random Variables .. Independent random Variables .. Order Statistics .. Sum of Two Independent random Variables: Discrete Case .. Sum of Two Independent random Variables: continuous Case Conditional Distributions: Discrete Case.

7 Conditional Distributions: continuous Case .. Joint Probability Distributions of Functions of random Vari-ables .. 46611 Properties of Expected Value of a Function of Two random Variables .. Covariance and Variance of Sums .. The Coefficient of Correlation .. Conditional Expectation .. Double Expectation .. Conditional Variance .. 51812 Moment Generating Functions and the Central Limit Moment Generating Functions .. Moment Generating Functions of Sums of Independent RVs . The Central Limit Theorem .. 544 Sample Exam 1553 Sample Exam 2567 Sample Exam 3583 Sample Exam 4597 Sample Exam 5611 Sample Exam 6625 Sample Exam 76394 CONTENTSS ample Exam 8653 Sample Exam 9667 Sample Exam 10681 Answer Keys697 Bibliography785 Index787 Chapter 1 Set Theory PrerequisiteTwo approaches of the concept of Probability will be introduced later in thebook: The classical Probability and the experimental Probability .

8 The formerapproach is developed using the foundation of set theory, and a quick reviewof the theory is in order. Readers familiar with the basics of set theory suchas set builder notation, Venn diagrams, and the basic operations on sets,(unions, intersections, and complements) can skip this the most basic term in Mathematics . Some synonyms of a set areclassorcollection. In this chapter, we introduce the concept of a set and itsvarious operations and then study the properties of these this book, we assume that the reader is familiar with the fol-lowing number systems and the algebraic operations and properties of suchsystems: The set of all positive integersN={1,2,3, }. the set of whole NumbersW={0,1,2,3, }.

9 The set of all integersZ={ , 3, 2, 1,0,1,2,3, }. The set of all rational numbersQ={ab:a,b Zwith b6= 0}. The setRof all real 1. SET THEORY Some Basic DefinitionsWe define asetas a collection ofwell-definedobjects (calledelementsormembers) such that for any given object one can assert without disputethat either the object is in the set or not but not both. Sets are usually willbe represented by upper case letters. When an objectxbelongs to a setA,we writex A,otherwise, we use the notationx6 , we mention herethat the members of a set can be sets of the following is a set.(a) The collection of good movies.(b) The collection of men 65 years of age in a certain (a) Answering a question about whether a movie is good or not may be sub-ject to dispute, the collection of good movies is not a well-defined set.

10 (b) This collection is a well-defined set since a man is either 65 years old ornotNext, we introduce a couple of set representations. The first one is to list,without repetition, the elements of the set. For example, ifAis the solutionset to the equationx2 4 = 0 thenA={ 2,2}.The order of how elementsof a set appear is irrelevant. We refer to this type of representation as thetabular formThe other way to represent a set is to describe a property that characterizesthe elements of the set. This is known as theset-builderrepresentation ofa set. For example, the setAabove can be written asA={x|xis an integersatisfyingx2 4 = 0}.The symbol|stands for the statement such as .We define theemptyset, denoted by ,to be the set with no elements.