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<p>A Probability Course for the ActuariesA Preparation for Exam P/1Marcel B. FinanArkansas Tech Universityc All Rights ReservedPreliminary DraftLast updatedMarch 20, 20122In memory of my parentsAugust 1, 2008January 7, 2009PrefaceThe present manuscript is designed mainly to help students prepare for theProbability Exam (Exam P/1), the rst actuarial examination administeredby the Society of Actuaries. This examination tests a students knowledge ofthe fundamental probability tools for quantitatively assessing risk. A thor-ough command of calculus is assumed.More information about the exam can be found on the webpage of the Soci-ety of Actuaries www.soa.org.Problems taken from samples of the Exam P/1 provided by the Society ofActuaries will be indicated by the symbol .The ow of topics in the book follows very closely that of Rosss A FirstCourse in Probability.This manuscript can be used for personal use or class use, but not for com-mercial purposes. If you nd any errors, I would appreciate hearing fromyou: mnan@atu.eduThis manuscript is also suitable for a one semester course in an undergradu-ate course in probability theory. Answer keys to text problems are found atthe end of the book.This project has been partially supported by a research grant from ArkansasTech University.Marcel B. FinanRussellville, ARMay 200734 PREFACEContentsPreface 3Basic Operations on Sets 91 Basic Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Counting and Combinatorics 313 The Fundamental Principle of Counting . . . . . . . . . . . . . . 314 Permutations and Combinations . . . . . . . . . . . . . . . . . . . 375 Permutations and Combinations with Indistinguishable Objects . 47Probability: Denitions and Properties 576 Basic Denitions and Axioms of Probability . . . . . . . . . . . . 577 Properties of Probability . . . . . . . . . . . . . . . . . . . . . . . 658 Probability and Counting Techniques . . . . . . . . . . . . . . . . 74Conditional Probability and Independence 819 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . 8110 Posterior Probabilities: Bayes Formula . . . . . . . . . . . . . . 8911 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . 10012 Odds and Conditional Probability . . . . . . . . . . . . . . . . . 109Discrete Random Variables 11313 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 11314 Probability Mass Function and Cumulative Distribution Function11915 Expected Value of a Discrete Random Variable . . . . . . . . . . 12716 Expected Value of a Function of a Discrete Random Variable . . 13517 Variance and Standard Deviation . . . . . . . . . . . . . . . . . 14256 CONTENTS18 Binomial and Multinomial Random Variables . . . . . . . . . . . 14819 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . 16220 Other Discrete Random Variables . . . . . . . . . . . . . . . . . 17220.1 Geometric Random Variable . . . . . . . . . . . . . . . . 17220.2 Negative Binomial Random Variable . . . . . . . . . . . . 17920.3 Hypergeometric Random Variable . . . . . . . . . . . . . 18721 Properties of the Cumulative Distribution Function . . . . . . . 193Continuous Random Variables 20722 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 20723 Expectation, Variance and Standard Deviation . . . . . . . . . . 22024 The Uniform Distribution Function . . . . . . . . . . . . . . . . 23825 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . 24326 Exponential Random Variables . . . . . . . . . . . . . . . . . . . 25827 Gamma and Beta Distributions . . . . . . . . . . . . . . . . . . 26828 The Distribution of a Function of a Random Variable . . . . . . 280Joint Distributions 28929 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 28930 Independent Random Variables . . . . . . . . . . . . . . . . . . 30331 Sum of Two Independent Random Variables . . . . . . . . . . . 31431.1 Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . 31431.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . . 31932 Conditional Distributions: Discrete Case . . . . . . . . . . . . . 32833 Conditional Distributions: Continuous Case . . . . . . . . . . . . 33534 Joint Probability Distributions of Functions of Random Variables 344Properties of Expectation 35135 Expected Value of a Function of Two Random Variables . . . . . 35136 Covariance, Variance of Sums, and Correlations . . . . . . . . . 36237 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 37638 Moment Generating Functions . . . . . . . . . . . . . . . . . . . 388Limit Theorems 40339 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . 40339.1 The Weak Law of Large Numbers . . . . . . . . . . . . . 40339.2 The Strong Law of Large Numbers . . . . . . . . . . . . . 40940 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 420CONTENTS 741 More Useful Probabilistic Inequalities . . . . . . . . . . . . . . . 430Appendix 43742 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 43743 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 44444 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . 45745 Risk Management and Insurance . . . . . . . . . . . . . . . . . . 462Answer Keys 471BIBLIOGRAPHY 5258 CONTENTSBasic Operations on SetsThe axiomatic approach to probability is developed using the foundation ofset theory, and a quick review of the theory is in order. If you are famil-iar with set builder notation, Venn diagrams, and the basic operations onsets, (unions, intersections, and complements), then you have a good starton what we will need right away from set theory.Set is the most basic term in mathematics. Some synonyms of a set areclass or collection. In this chapter we introduce the concept of a set and itsvarious operations and then study the properties of these operations.Throughout this book, we assume that the reader is familiar with the follow-ing number systems: The set of all positive integersN = {1, 2, 3, }. The set of all integersZ = { , 3, 2, 1, 0, 1, 2, 3, }. The set of all rational numbersQ = {ab : a, b Z with b = 0}. The set R of all real numbers.1 Basic DenitionsWe dene a set A as a collection of well-dened objects (called elements ormembers of A) such that for any given object x either one (but not both)910 BASIC OPERATIONS ON SETSof the following holds: x belongs to A and we write x A. x does not belong to A, and in this case we write x A.Example 1.1Which of the following is a well-dened set.(a) The collection of good books.(b) The collection of left-handed individuals in Russellville.Solution.(a) The collection of good books is not a well-dened set since the answer tothe question Is My Life a good book? may be subject to dispute.(b) This collection is a well-dened set since a person is either left-handed orright-handed. Of course, we are ignoring those few who can use both handsThere are two dierent ways to represent a set. The rst one is to list,without repetition, the elements of the set. For example, if A is the solutionset to the equation x24 = 0 then A = {2, 2}. The other way to representa set is to describe a property that characterizes the elements of the set. Thisis known as the set-builder representation of a set. For example, the set Aabove can be written as A = {x|x is an integer satisfying x24 = 0}.We dene the empty set, denoted by , to be the set with no elements. Aset which is not empty is called a nonempty set.Example 1.2List the elements of the following sets.(a) {x|x is a real number such that x2= 1}.(b) {x|x is an integer such that x23 = 0}.Solution.(a) {1, 1}.(b) Since the only solutions to the given equation are 3 and3 and bothare not integers, the set in question is the empty setExample 1.3Use a property to give a description of each of the following sets.(a) {a, e, i, o, u}.(b) {1, 3, 5, 7, 9}.1 BASIC DEFINITIONS 11Solution.(a) {x|x is a vowel}.(b) {n N|n is odd and less than 10 }The rst arithmetic operation involving sets that we consider is the equalityof two sets. Two sets A and B are said to be equal if and only if they containthe same elements. We write A = B. For non-equal sets we write A = B. Inthis case, the two sets do not contain the same elements.Example 1.4Determine whether each of the following pairs of sets are equal.(a) {1, 3, 5} and {5, 3, 1}.(b) {{1}} and {1, {1}}.Solution.(a) Since the order of listing elements in a set is irrelevant, {1, 3, 5} ={5, 3, 1}.(b) Since one of the sets has exactly one member and the other has two,{{1}} = {1, {1}}In set theory, the number of elements in a set has a special name. It iscalled the cardinality of the set. We write n(A) to denote the cardinality ofthe set A. If A has a nite cardinality we say that A is a nite set. Other-wise, it is called innite. For innite set, we write n(A) = . For example,n(N) = .Can two innite sets have the same cardinality? The answer is yes. If A andB are two sets (nite or innite) and there is a bijection from A to B (i.e.a one-to-one and onto function) then the two sets are said to have the samecardinality, i.e. n(A) = n(B).Example 1.5What is the cardinality of each of the following sets?(a) .(b) {}.(c) {a, {a}, {a, {a}}}.Solution.(a) n() = 0.12 BASIC OPERATIONS ON SETS(b) This is a set consisting of one element . Thus, n({}) = 1.(c) n({a, {a}, {a, {a}}}) = 3Now, one compares numbers using inequalities. The corresponding notionfor sets is the concept of a subset: Let A and B be two sets. We say thatA is a subset of B, denoted by A B, if and only if every element of A isalso an element of B. If there exists an element of A which is not in B thenwe write A B.For any set A we have A A. That is, every set has at least two subsets.Also, keep in mind that the empty set is a subset of any set.Example 1.6Suppose that A = {2, 4, 6}, B = {2, 6}, and C = {4, 6}. Determine which ofthese sets are subsets of which other of these sets.Solution.B A and C AIf sets A and B are represented as regions in the plane, relationships be-tween A and B can be represented by pictures, called Venn diagrams.Example 1.7Represent A B C using Venn diagram.Solution.The Venn diagram is given in Figure 1.1Figure 1.1Let A and B be two sets. We say that A is a proper subset of B, denotedby A B, if A B and A = B. Thus, to show that A is a proper subset ofB we must show that every element of A is an element of B and there is anelement of B which is not in A.1 BASIC DEFINITIONS 13Example 1.8Order the sets of numbers: Z, R, Q, N using Solution.N Z Q RExample 1.9Determine whether each of the following statements is true or false.(a) x {x} (b) {x} {x} (c) {x} {x}(d) {x} {{x}} (e) {x} (f) {x}Solution.(a) True (b) True (c) False since {x} is a set consisting of a single element xand so {x} is not a member of this set (d) True (e) True (f) False since {x}does not have as a listed memberNow, the collection of all subsets of a set A is of importance. We denotethis set by P(A) and we call it the power set of A.Example 1.10Find the power set of A = {a, b, c}.Solution.P(A) = {, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}We conclude this section, by introducing the concept of mathematical induc-tion: We want to prove that some statement P(n) is true for any nonnegativeinteger n n0. The steps of mathematical induction are as follows:(i) (Basis of induction) Show that P(n0) is true.(ii) (Induction hypothesis) Assume P(n0), P(n0 + 1), , P(n) are true.(iii) (Induction step) Show that P(n + 1) is true.Example 1.11(a) Use induction to show that if n(A) = n then n(P(A)) = 2n, where n 0and n N.(b) If P(A) has 256 elements, how many elements are there in A?14 BASIC OPERATIONS ON SETSSolution.(a) We apply induction to prove the claim. If n = 0 then A = and inthis...</p>