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Lecture Notes in Actuarial Mathematics A Probability Course for the Actuaries A Preparation for Exam P/1 Marcel B. Finan
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Preface The present manuscript is designed mainly to help students prepare for the Probability Exam (known as Exam P/1), the first actuarial examination ad- ministered 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 Soci- ety of Actuaries www.soa.org. Problems taken from previous exams provided by the Society of Actuaries will be indicated by the symbol ‡. This manuscript can be used for personal use or class use, but not for com- mercial 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 undergradu- ate course in probability theory. Answer keys to text problems are found at the end of the book. Marcel B. Finan Russellville, AR May, 2008 i
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Contents Preface i Set Theory Prerequisite 5 1 Some Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Counting and Combinatorics 31 3 The Fundamental Principle of Counting . . . . . . . . . . . . . . 31 4 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Probability: Definitions and Properties 55 6 Sample Space, Events, Probability Measure . . . . . . . . . . . . 55 7 Probability of Intersection, Union, and Complementary Event . . 67 8 Probability and Counting Techniques . . . . . . . . . . . . . . . . 78 Conditional Probability and Independence 87 9 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . 87 10 Posterior Probabilities: Bayes’ Formula . . . . . . . . . . . . . . 97 11 Independent Events . . . . . . . . . . . . . . . . . . . . . . . . . 110 12 Odds and Conditional Probability . . . . . . . . . . . . . . . . . 121 Discrete Random Variables 125 13 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 125 14 Probability Mass Function and Cumulative Distribution Function133 15 Expected Value of a Discrete Random Variable . . . . . . . . . . 143 16 Expected Value of a Function of a Discrete Random Variable . . 153 17 Variance and Standard Deviation of a Discrete Random Variable 162 1
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2 CONTENTS Commonly Used Discrete Random Variables 171 18 Uniform Discrete Random Variable . . . . . . . . . . . . . . . . 171 19 Bernoulli Trials and Binomial Distributions . . . . . . . . . . . . 177 20 The Expected Value and Variance of the Binomial Distribution . 185 21 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . 194 22 Geometric Random Variable . . . . . . . . . . . . . . . . . . . . 207 23 Negative Binomial Random Variable . . . . . . . . . . . . . . . . 216 24 Hyper-geometric Random Variable . . . . . . . . . . . . . . . . . 224 Cumulative and Survival Distribution Functions 233 25 The Cumulative Distribution Function . . . . . . . . . . . . . . 233 26 The Survival Distribution Function . . . . . . . . . . . . . . . . 248 Calculus Prerequisite I 255 27 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Continuous Random Variables 267 28 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 267 29 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . 282 30 Median, Mode, and Percentiles . . . . . . . . . . . . . . . . . . . 296 31 The Continuous Uniform Distribution Function . . . . . . . . . . 304 32 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . 312 33 The Normal Approximation to the Binomial Distribution . . . . 323 34 Exponential Random Variables . . . . . . . . . . . . . . . . . . . 328 35 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . 338 36 The Distribution of a Function of a Continuous Random Variable 345 Calculus Prerequisite II 353 37 Graphing Systems of Inequalities in Two Variables . . . . . . . . 353 38 Iterated Double Integrals . . . . . . . . . . . . . . . . . . . . . . 359 Joint Distributions 369 39 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 369 40 Independent Random Variables . . . . . . . . . . . . . . . . . . 385 41 Sum of Two Independent Random Variables: Discrete Case . . . 398 42 Sum of Two Independent Random Variables: Continuous Case . 405 43 Conditional Distributions: Discrete Case . . . . . . . . . . . . . 415 44 Conditional Distributions: Continuous Case . . . . . . . . . . . . 425
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CONTENTS 3 45 Joint Probability Distributions of Functions of Random Variables 435 Properties of Expectation 443 46 Expected Value of a Function of Two Random Variables . . . . . 443 47 Covariance and Variance of Sums . . . . . . . . . . . . . . . . . 456 48 The Coefficient of Correlation . . . . . . . . . . . . . . . . . . . 464 49 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 471 50 Double Expectation . . . . . . . . . . . . . . . . . . . . . . . . . 480 51 Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . . 490 Moment Generating Functions and the Central Limit Theorem499 52 Moment Generating Functions . . . . . . . . . . . . . . . . . . . 499 53 Moment Generating Functions of Sums of Independent RVs . . . 507 54 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 517 Sample Exam 1 525 Sample Exam 2 539 Sample Exam 3 555 Sample Exam 4 569 Sample Exam 5 583 Sample Exam 6 597 Sample Exam 7 611 Sample Exam 8 625 Sample Exam 9 639 Sample Exam 10 653 Answer Keys 669 Bibliography 757 Index 759
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Set Theory Prerequisite Two approaches of the concept of probability will be introduced later in the book: The classical probability and the experimental probability. The former approach is developed using the foundation of set theory, and a quick review of the theory is in order. Readers familiar with the basics of set theory such as set builder notation, Venn diagrams, and the basic operations on sets, (unions, intersections, and complements) can skip this chapter. However, going through the content of this chapter will provide a good start on what we will need right away from set theory. Set is the most basic term in mathematics. Some synonyms of a set are class or collection. In this chapter, we introduce the concept of a set and its various operations and then study the properties of these operations. Throughout this book, we assume that the reader is familiar with the fol- lowing number systems and the algebraic operations and properties of such systems: • The set of all positive integers N = {1,2,3,· · · }. • The set of all integers Z = {· · · ,−3,−2,−1,0,1,2,3,· · · }. • The set of all rational numbers a Q = { : a, b ∈ Z with b 6= 0}. b • The set R of all real numbers. 5
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6 SET THEORY PREREQUISITE 1 Some Basic Definitions We define a set as a collection of well-defined objects (called elements or members of A) such that for any given object one can assert without dispute that either the object is in the set or not but not both. Sets are usually will be represented by upper case letters. When an object x belongs to a set A, we write x ∈ A, otherwise, we use the notation x 6∈ A. Also, we mention here that the members of a set can be sets themselves. Example 1.1 Which of the following is a set. (a) The collection of good movies. (b) The collection of right-handed individuals in Dallas. Exclude mixed- handedness. Solution. (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. (b) This collection is a well-defined set since a person is either left-handed or right-handed Next, we introduce a couple of set representations. The first one is to list, without repetition, the elements of the set. For example, if A is the solution set to the equation x2 −4 = 0 then A = {−2,2}. The other way to represent a set is to describe a property that characterizes the elements of the set. This is known as the set-builder representation of a set. For example, the set A above can be written as A = {x|x is an integer satisfying x2 − 4 = 0}. We define the empty set, denoted by ∅, to be the set with no elements. A set which is not empty is called a non-empty set. Example 1.2 List 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 x2 − 3 = 0}. Solution. (a) {−1,1}. √ √ (b) Since the only solutions to the given equation are − 3 and 3 and both are not integers, the set in question is the empty set
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1 SOME BASIC DEFINITIONS 7 Example 1.3 Use a property characterizing the members of the following sets. (a) {a,e,i,o,u}. (b) {1,3,5,7,9}. Solution. (a) {x|x is a vowel}. (b) {n|n ∈ N is odd and less than 10 } The first arithmetic operation involving sets that we consider is the equality of two sets. Two sets A and B are said to be equal if and only if they contain the same elements. We write A = B. For non-equal sets we write A 6= B. In this case, the two sets do not contain the same elements. Example 1.4 Determine 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}} 6= {1,{1}} In set theory, the number of elements in a set has a special name. It is called the cardinality of the set. We write #(A) to denote the cardinality of the set A. If A has a finite cardinality we say that A is a finite set. Oth- erwise, it is called infinite. For example, N is an infinite set. Can two infinite sets have the same cardinality? The answer is yes. If A and B are two sets (finite or infinite) and there is a bijection from A to B( i.e., a one-to-one1 and onto2 function) then the two sets are said to have the same cardinality and we write #(A) = #(B). If #(A) is either finite or has the same cardinality as N then we say that A is countable. A set that is not countable is said to be uncountable. 1A function f : A 7−→ B is a one-to-one function if f(m) = f(n) implies m = n, where m, n ∈ A. 2A function f : A 7−→ B is an onto function if for every b ∈ B, there is an a ∈ A such that b = f(a).
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8 SET THEORY PREREQUISITE Example 1.5 What is the cardinality of each of the following sets? (a) ∅. (b) {∅}. (c) A = {a,{a},{a,{a}}}. Solution. (a) #(∅) = 0. (b) This is a set consisting of one element ∅. Thus, #({∅}) = 1. (c) #(A) = 3 Example 1.6 (a) Show that the set A = {a1, a2.· · · , an,· · · } is countable. (b) Let A be the set of all infinite sequences of the digits 0 and 1. Show that A is uncountable. Solution. (a) We first show that the map f : N 7−→ A defined by f(n) = an is one-to- one. Indeed, if f(n) = f(m) then an = am and this implies n = m. Now, any member in A is of the form an = f(n). Thus, A is onto. It follows that A is countable. (b) We will argue by contradiction. Suppose that A is countable with ele- ments a1, a2,· · · . where each ai is an infinite sequence of the digits 0 and 1. Let a be the infinite sequence with the first digit of 0 or 1 different from the first digit of a1, the second digit of 0 or 1 different from the second digit of a ,· · · , the nth digit is different from the nth digit of a , etc. Thus, a is 2 n an infinite sequence of the digits 0 and 1 which is not in A, a contradiction. Hence, A is uncountable Now, let A and B be two sets. We say that A is a subset of B, denoted by A ⊆ B, if and only if every element of A is also an element of B. If there exists an element of A which is not in B, then we write A 6⊆ B. A universal set U is the collection of all objects in a particular context or theory. All other sets in that framework constitute subsets of the universal set. 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 1.7 Suppose that A = {2,4,6}, B = {2,6}, and C = {4,6}. Determine which of these sets are subsets of which other of these sets.
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1 SOME BASIC DEFINITIONS 9 Solution. Clearly, B ⊆ A and C ⊆ A Subsets of a universal set can be represented by circles or closed curves with a rectangle that represents the universal set. We refer to such pictorial rep- resentation as a Venn diagram. Example 1.8 Represent A ⊆ B ⊆ C using Venn diagram. Solution. The Venn diagram is given in Figure 1.1 Figure 1.1 Next, let A and B be two sets. We say that A is a proper subset of B, denoted by A ⊂ B, if A ⊆ B and A 6= B. Thus, to show that A is a proper subset of B, we must show that every element of A is an element of B and there is an element of B which is not in A. Example 1.9 Order the sets of numbers: Z,R,Q,N using ⊂ Solution. We have, N ⊂ Z ⊂ Q ⊂ R Example 1.10 Determine 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}
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10 SET THEORY PREREQUISITE Solution. (a) True (b) True (c) False since {x} is a set consisting of a single element x and so {x} is not a member of this set (d) True (e) True (f) False since {x} does not have ∅ as a listed member Now, the collection of all subsets of a set A is of importance. We denote this set by P(A) and we call it the power set of A. Example 1.11 Find 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 in- duction: We want to prove that some statement P(n) is true for any non- negative integer 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 1.12 (a) Use induction to show that if #(A) = n then #(P(A)) = 2n, where n ≥ 0 and n ∈ N. (b) If P(A) has 256 elements, how many elements are there in A? Solution. (a) We apply induction to prove the claim. If n = 0 then A = ∅ and in this case P(A) = {∅}. Thus, #(P(A)) = 1 = 20. As induction hy- pothesis, suppose that if #(A) = k, where k = n0, n0 + 1,· · · , n, then n(P(A)) = 2k. Let B = {a , a ,· · · , a , a }. Then P(B) consists of all 1 2 n n+1 subsets of {a1, a2,· · · , an} together with all subsets of {a1, a2,· · · , an} with the element a added to them. Hence, #(P(B)) = 2n + 2n = 2·2n = 2n+1. n+1 (b) Since #(P(A)) = 256 = 28, by (a) we have #(A) = 8 Example 1.13 ∑n Use induction to show that (2i − 1) = n2, n ∈ N. i=1
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1 SOME BASIC DEFINITIONS 11 Solution. ∑1 If n = 1 we have 12 = 2(1) − 1 = (2i − 1). Suppose that the result i=1 is true for k = 1,2,· · · , n. We will show that it is true for n + 1. Indeed, ∑n+1 ∑n (2i − 1) = (2i − 1) + 2(n + 1) − 1 = n2 + 2n + 2 − 1 = (n + 1)2 i=1 i=1
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12 SET THEORY PREREQUISITE Practice Problems Problem 1.1 Consider the experiment of rolling a die. List the elements of the set A = {x|x shows a face with prime number}. Recall that a prime number is a positive integer greater than 1 with only two different divisors: 1 and the number itself. Problem 1.2 Consider the random experiment of tossing a coin three times. (a) Let S be the collection of all outcomes of this experiment. List the ele- ments of S. Use H for head and T for tail. (b) Let E be the subset of S with more than one tail. List the elements of E. (c) Suppose F = {T HH, HT H, HHT, HHH}. Write F in set-builder nota- tion. Problem 1.3 Consider the experiment of tossing a coin three times. Let E be the collection of outcomes with at least one head and F the collection of outcomes of more than one head. Compare the two sets E and F. Problem 1.4 A hand of 5 cards is dealt from a deck of 52 cards. Let E be the event that the hand contains 5 aces. List the elements of E. Problem 1.5 Prove the following properties: (a) Reflexive Property: A ⊆ A. (b) Antisymmetric Property: If A ⊆ B and B ⊆ A then A = B. (c) Transitive Property: If A ⊆ B and B ⊆ C then A ⊆ C. Problem 1.6 Prove by using mathematical induction that n(n + 1) 1 + 2 + 3 + · · · + n = , n ∈ N. 2
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1 SOME BASIC DEFINITIONS 13 Problem 1.7 Prove by using mathematical induction that 2 2 2 2 n(n + 1)(2n + 1) 1 + 2 + 3 + · · · + n = , n ∈ N. 6 Problem 1.8 Use induction to show that (1 + x)n ≥ 1 + nx for all non-negative integer n, where x > −1. Problem 1.9 Use induction to show that 1 − an 1 + a + a2 + · · · + an−1 = , a 6= 1. 1 − a Problem 1.10 Subway prepared 60 4-inch sandwiches for a birthday party. Among these sandwiches, 45 of them had tomatoes, 30 had both tomatoes and onions, and 5 had neither tomatoes nor onions. Using a Venn diagram, how many sandwiches did he make with (a) tomatoes or onions? (b) onions? (c) onions but not tomatoes? Problem 1.11 A camp of international students has 110 students. Among these students, 75 speak English, 52 speak Spanish, 50 speak French, 33 speak English and Spanish, 30 speak English and French, 22 speak Spanish and French, 13 speak all three languages. How many students speak (a) English and Spanish, but not French, (b) neither English, Spanish, nor French, (c) French, but neither English nor Spanish, (d) English, but not Spanish, (e) only one of the three languages, (f) exactly two of the three languages.
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14 SET THEORY PREREQUISITE Problem 1.12 An experiment consists of the following two stages: (1) a fair coin is tossed (2) if the coin shows a head, then a fair die is rolled; otherwise, the coin is flipped again. An outcome of this experiment is a pair of the form (outcome from stage 1, outcome from stage 2). Let S be the collection of all outcomes. List the elements of S and then find the cardinality of S. Problem 1.13 Show that the function f : R 7−→ R defined by f(x) = 3x + 5 is one-to-one and onto. Problem 1.14 Find #(A) if #(P(A)) = 32. Problem 1.15 Consider the function f : N 7−→ Z defined by { n, if n is even f(n) = −n2−1, if n is odd. 2 (a) Show that f(n) = f(m) only if n and m have the same parity, i.e., either both are even or both are odd.. (b) Show that Z is countable. Problem 1.16 Let A be a non-empty set and f : A 7−→ P(A) be any function. Let B = {a ∈ A|a 6∈ f(a)}. Clearly, B ∈ P(A). Show that there is no b ∈ A such that f(b) = B. Hence, there is no onto map from A to P(A). Problem 1.17 Use the previous problem to show that P(N) is uncountable. Problem 1.18 Show that the sets (0,∞) and R have the same cardinality. Problem 1.19 Show that the function f : N × N 7−→ N defined by f(n, m) = 2n3m is one-to-one.
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1 SOME BASIC DEFINITIONS 15 Problem 1.20 A marketing survey indicates that 60% of the population owns a laptop, 30% owns a desktop computer, and 20% owns both a laptop and a desktop computers. What percent of the population owns a laptop or a desktop, but not both?
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16 SET THEORY PREREQUISITE 2 Set Operations In this section we introduce various operations on sets and study the prop- erties of these operations. Complements If U is a given set whose subsets are under consideration, then we call U a universal set. Let U be a universal set and A, B be two subsets of U. The absolute complement of A (See Figure 2.1(I)) is the set Ac = {x ∈ U|x 6∈ A}. Example 2.1 Find the complement of A = {1,2,3} if U = {1,2,3,4,5,6}. Solution. From the definition, Ac = {4,5,6} The relative complement of A with respect to B (See Figure 2.1(II)) is the set B − A = {x ∈ U|x ∈ B and x 6∈ A}. Figure 2.1 Example 2.2 Let A = {1,2,3} and B = {{1,2},3}. Find A − B. Solution. The elements of A that are not in B are 1 and 2. Thus, A − B = {1,2}
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