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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

March 20, 2012

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 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 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 of

Actuaries will be indicated by the symbolz:

The ow of topics in the book follows very closely that of Ross'sA First

Course 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 from you: mnan@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. This project has been partially supported by a research grant from Arkansas

Tech University.

Marcel B. Finan

Russellville, AR

May 2007

3

4PREFACE

Contents

Preface 3

Basic Operations on Sets 9

1 Basic Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 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: Denitions and Properties 57

6 Basic Denitions 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

6CONTENTS

18 Binomial and Multinomial Random Variables . . . . . . . . . . . 148

19 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . 162

20 Other Discrete Random Variables . . . . . . . . . . . . . . . . . 172

20.1 Geometric Random Variable . . . . . . . . . . . . . . . . 172

20.2 Negative Binomial Random Variable . . . . . . . . . . . . 179

20.3 Hypergeometric Random Variable . . . . . . . . . . . . . 187

21 Properties of the Cumulative Distribution Function . . . . . . . 193

Continuous Random Variables 207

22 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 207

23 Expectation, Variance and Standard Deviation . . . . . . . . . . 220

24 The Uniform Distribution Function . . . . . . . . . . . . . . . . 238

25 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . 243

26 Exponential Random Variables . . . . . . . . . . . . . . . . . . . 258

27 Gamma and Beta Distributions . . . . . . . . . . . . . . . . . . 268

28 The Distribution of a Function of a Random Variable . . . . . . 280

Joint Distributions 289

29 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 289

30 Independent Random Variables . . . . . . . . . . . . . . . . . . 303

31 Sum of Two Independent Random Variables . . . . . . . . . . . 314

31.1 Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . 314

31.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . . 319

32 Conditional Distributions: Discrete Case . . . . . . . . . . . . . 328

33 Conditional Distributions: Continuous Case . . . . . . . . . . . . 335

34 Joint Probability Distributions of Functions of Random Variables344

Properties of Expectation 351

35 Expected Value of a Function of Two Random Variables . . . . . 351

36 Covariance, Variance of Sums, and Correlations . . . . . . . . . 362

37 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 376

38 Moment Generating Functions . . . . . . . . . . . . . . . . . . . 388

Limit Theorems 403

39 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . 403

39.1 The Weak Law of Large Numbers . . . . . . . . . . . . . 403

39.2 The Strong Law of Large Numbers . . . . . . . . . . . . . 409

40 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 420

CONTENTS7

41 More Useful Probabilistic Inequalities . . . . . . . . . . . . . . . 430

Appendix 437

42 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 437

43 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

44 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . 457

45 Risk Management and Insurance . . . . . . . . . . . . . . . . . . 462

Answer Keys 471

BIBLIOGRAPHY 525

8CONTENTS

Basic Operations on Sets

The axiomatic approach to probability is developed using the foundation of set 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 on sets, (unions, intersections, and complements), then you have a good start on what we will need right away from set theory. Setis the most basic term in mathematics. Some synonyms of a set are classorcollection. 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 follow- ing number systems:

The set of all positive integers

N=f1;2;3;g:

The set of all integers

Z=f;3;2;1;0;1;2;3;g:

The set of all rational numbers

Q=fab :a;b2Zwith b6= 0g:

The setRof all real numbers.

1 Basic Denitions

We dene asetAas a collection ofwell-denedobjects (calledelementsor membersofA) such that for any given objectxeither one (but not both) 9

10BASIC OPERATIONS ON SETS

of the following holds: xbelongs toAand we writex2A: xdoes not belong toA;and in this case we writex62A:

Example 1.1

Which 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 to the question \IsMy Lifea good book?" may be subject to dispute. (b) This collection is a well-dened set since a person is either left-handed or

right-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, ifAis the solution set to the equationx24 = 0 thenA=f2;2g:The other way to represent a set is to describe a property that characterizes the elements of the set. This is known as theset-builderrepresentation of a set. For example, the setA above can be written asA=fxjxis an integer satisfyingx24 = 0g: We dene theemptyset, denoted by;;to be the set with no elements. A set which is not empty is called anonemptyset.

Example 1.2

List the elements of the following sets.

(a)fxjxis a real number such thatx2= 1g: (b)fxjxis an integer such thatx23 = 0g:

Solution.

(a)f1;1g: (b) Since the only solutions to the given equation arep3 and p3 and both are not integers, the set in question is the empty setExample 1.3 Use a property to give a description of each of the following sets. (a)fa;e;i;o;ug: (b)f1;3;5;7;9g:

1 BASIC DEFINITIONS11

Solution.

(a)fxjxis a vowelg.

(b)fn2Njnis odd and less than 10gThe rst arithmetic operation involving sets that we consider is the equality

of two sets. Two setsAandBare said to beequalif and only if they contain the same elements. We writeA=B:For non-equal sets we writeA6=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)f1;3;5gandf5;3;1g: (b)ff1ggandf1;f1gg:

Solution.

(a) Since the order of listing elements in a set is irrelevant,f1;3;5g= f5;3;1g: (b) Since one of the sets has exactly one member and the other has two, ff1gg 6=f1;f1ggIn set theory, the number of elements in a set has a special name. It is called thecardinalityof the set. We writen(A) to denote the cardinality of the setA:IfAhas a nite cardinality we say thatAis aniteset. Other- wise, it is calledinnite. For innite set, we writen(A) =1:For example, n(N) =1: Can two innite sets have the same cardinality? The answer is yes. IfAand Bare two sets (nite or innite) and there is a bijection fromAtoB(i.e. a one-to-one and onto function) then the two sets are said to have the same cardinality, i.e.n(A) =n(B):

Example 1.5

What is the cardinality of each of the following sets? (a);: (b)f;g: (c)fa;fag;fa;faggg:

Solution.

(a)n(;) = 0:

12BASIC OPERATIONS ON SETS

(b) This is a set consisting of one element;:Thus,n(f;g) = 1: (c)n(fa;fag;fa;faggg) = 3Now, one compares numbers using inequalities. The corresponding notion for sets is the concept of a subset: LetAandBbe two sets. We say that Ais asubsetofB, denoted byAB;if and only if every element ofAis also an element ofB:If there exists an element ofAwhich is not inBthen we writeA6B: For any setAwe have; AA:That is, every set has at least two subsets. Also, keep in mind that the empty set is a subset of any set.

Example 1.6

Suppose thatA=f2;4;6g; B=f2;6g;andC=f4;6g:Determine which of these sets are subsets of which other of these sets.

Solution.

BAandCAIf setsAandBare represented as regions in the plane, relationships be- tweenAandBcan be represented by pictures, calledVenn diagrams.

Example 1.7

RepresentABCusing Venn diagram.

Solution.

The Venn diagram is given in Figure 1.1Figure 1.1

LetAandBbe two sets. We say thatAis apropersubset ofB;denoted byAB;ifABandA6=B:Thus, to show thatAis a proper subset of Bwe must show that every element ofAis an element ofBand there is an element ofBwhich is not inA:

1 BASIC DEFINITIONS13

Example 1.8

Order the sets of numbers:Z;R;Q;Nusing

Solution.

NZQRExample 1.9

Determine whether each of the following statements is true or false. (a)x2 fxg(b)fxg fxg(c)fxg 2 fxg (d)fxg 2 ffxgg(e); fxg(f); 2 fxg

Solution.

(a) True (b) True (c) False sincefxgis a set consisting of a single elementx and sofxgis not a member of this set (d) True (e) True (f) False sincefxg

does not have;as a listed memberNow, the collection of all subsets of a setAis of importance. We denote

this set byP(A) and we call it thepower setofA:

Example 1.10

Find the power set ofA=fa;b;cg:

Solution.

P(A) =f;;fag;fbg;fcg;fa;bg;fa;cg;fb;cg;fa;b;cggWe conclude this section, by introducing the concept of mathematical induc-

tion: We want to prove that some statementP(n) is true for any nonnegative integernn0:The steps of mathematical induction are as follows: (i) (Basis of induction) Show thatP(n0) is true. (ii) (Induction hypothesis) AssumeP(n0);P(n0+ 1);;P(n) are true. (iii) (Induction step) Show thatP(n+ 1) is true.

Example 1.11

(a) Use induction to show that ifn(A) =nthenn(P(A)) = 2n;wheren0 andn2N: (b) IfP(A) has 256 elements, how many elements are there inA?

14BASIC OPERATIONS ON SETS

Solution.

(a) We apply induction to prove the claim. Ifn= 0 thenA=;and in this caseP(A) =f;g:Thus,n(P(A)) = 1 = 20:As induction hypothesis, suppose that ifn(A) =nthenn(P(A)) = 2n:LetB=fa1;a2;;an;an+1g: ThenP(B) consists of all subsets offa1;a2;;angtogether with all subsets offa1;a2;;angwith the elementan+1added to them. Hence,n(P(B)) = 2 n+ 2n= 22n= 2n+1: (b) Sincen(P(A)) = 256 = 28;by (a) we haven(A) = 8Example 1.12

Use induction to show thatPn

i=1(2i1) =n2; n2N:

Solution.

Ifn= 1 we have 12= 2(1)1 =P1

i=1(2i1):Suppose that the result is true for up ton:We will show that it is true forn+ 1:Indeed,Pn+1 i=1(2i1) =Pn i=1(2i1) + 2(n+ 1)1 =n2+ 2n+ 21 = (n+ 1)2

1 BASIC DEFINITIONS15

Practice Problems

Problem 1.1

Consider the experiment of rolling a die. List the elements of the setA= fx:xshows a face with prime numberg. Recall that a prime number is a number with only two divisors: 1 and the number itself.

Problem 1.2

Consider the random experiment of tossing a coin three times. (a) LetSbe the collection of all outcomes of this experiment. List the ele- ments ofS:UseHfor head andTfor tail. (b) LetEbe the subset ofSwith more than one tail. List the elements of E. (c) SupposeF=fTHH;HTH;HHT;HHHg:WriteFin set-builder nota- tion.

Problem 1.3

Consider the experiment of tossing a coin three times. LetEbe the collection of outcomes with at least one head andFthe collection of outcomes of more than one head. Compare the two setsEandF:

Problem 1.4

A hand of 5 cards is dealt from a deck. LetEbe the event that the hand contains 5 aces. List the elements ofE:

Problem 1.5

Prove the following properties:

(a) Re exive Property:AA: (b) Antisymmetric Property: IfABandBAthenA=B: (c) Transitive Property: IfABandBCthenAC:

Problem 1.6

Prove by using mathematical induction that

1 + 2 + 3 ++n=n(n+ 1)2

; n2N:

Problem 1.7

Prove by using mathematical induction that

1

2+ 22+ 32++n2=n(n+ 1)(2n+ 1)6

; n2N:

16BASIC OPERATIONS ON SETS

Problem 1.8

Use induction to show that (1 +x)n1 +nxfor alln2N;wherex >1:

Problem 1.9

A caterer prepared 60 beef tacos for a birthday party. Among these tacos, he made 45 with tomatoes, 30 with both tomatoes and onions, and 5 with neither tomatoes nor onions. Using a Venn diagram, how many tacos did he make with (a) tomatoes or onions? (b) onions? (c) onions but not tomatoes?

Problem 1.10

A dormitory of college freshmen has 110 students. Among these students,

75 are taking English,

52 are taking history,

50 are taking math,

33 are taking English and history,

30 are taking English and math,

22 are taking history and math,

13 are taking English, history, and math.

How many students are taking

(a) English and history, but not math, (b) neither English, history, nor math, (c) math, but neither English nor history, (d) English, but not history, (e) only one of the three subjects, (f) exactly two of the three subjects.

Problem 1.11

An experiment consists of the following two stages: (1) rst a fair die is rolled (2) if the number appearing is even, then a fair coin is tossed; if the number appearing is odd, then the die is tossed again. An outcome of this experiment is a pair of the form (outcome from stage 1, outcome from stage

2). LetSbe the collection of all outcomes. List the elements ofSand then

nd the cardinality ofS:

2 SET OPERATIONS17

2 Set Operations

In this section we introduce various operations on sets and study the prop- erties of these operations.

Complements

IfUis a given set whose subsets are under consideration, then we callUa universal set.LetUbe a universal set andA;Bbe two subsets ofU:The absolute complementofA(See Figure 2.1(I)) is the set A c=fx2Ujx62Ag:

Example 2.1

Find the complement ofA=f1;2;3gifU=f1;2;3;4;5;6g:

Solution.

From the denition,Ac=f4;5;6gTherelative complementofAwith respect toB(See Figure 2.1(II)) is the set

BA=fx2Ujx2B and x62Ag:Figure 2.1

Example 2.2

LetA=f1;2;3gandB=ff1;2g;3g:FindAB:

Solution.

The elements ofAthat are not inBare 1 and 2. That is,AB=f1;2gUnion and Intersection

Given two setsAandB:TheunionofAandBis the set

A[B=fxjx2A or x2Bg

18BASIC OPERATIONS ON SETS

where the `or' is inclusive.(See Figure 2.2(a))Figure 2.2 The above denition can be extended to more than two sets. More precisely, ifA1;A2;;are sets then

1n=1An=fxjx2Aifor some i2Ng:

TheintersectionofAandBis the set (See Figure 2.2(b))

A\B=fxjx2A and x2Bg:

Example 2.3

Express each of the following events in terms of the eventsA;B;and C as well as the operations of complementation, union and intersection: (a) at least one of the eventsA;B;Coccurs; (b) at most one of the eventsA;B;Coccurs; (c) none of the eventsA;B;Coccurs; (d) all three eventsA;B;Coccur; (e) exactly one of the eventsA;B;Coccurs; (f) eventsAandBoccur, but notC; (g) either eventAoccurs or, if not, thenBalso does not occur.

In each case draw the corresponding Venn diagram.

Solution.

(a)A[B[C (b) (A\Bc\Cc)[(Ac\B\Cc)[(Ac\Bc\C)[(Ac\Bc\Cc) (c) (A[B[C)c=Ac\Bc\Cc (d)A\B\C (e) (A\Bc\Cc)[(Ac\B\Cc)[(Ac\Bc\C) (f)A\B\Cc

2 SET OPERATIONS19

(g)A[(Ac\Bc)Example 2.4 Translate the following set-theoretic notation into event language. For ex- ample, \A[B" means \AorBoccurs". (a)A\B (b)AB (c)A[BA\B (d)A(B[C) (e)AB (f)A\B=;

Solution.

(a)AandBoccur (b)Aoccurs andBdoes not occur (c)AorB;but not both, occur (d)Aoccurs, andBandCdo not occur (e) ifAoccurs, thenBoccurs but ifBoccurs thenAneed not accur. (f) ifAoccurs, thenBdoes not occur or ifBoccurs thenAdoes not occurExample 2.5 Find a simpler expression of [(A[B)\(A[C)\(Bc\Cc)] assuming all three sets intersect.

20BASIC OPERATIONS ON SETS

Solution.

Using a Venn diagram one can easily see that [(A[B)\(A[C)\(Bc\Cc)] =

A[A\(B[C)]IfA\B=;we say thatAandBaredisjointsets.

Example 2.6

LetAandBbe two non-empty sets. WriteAas the union of two disjoint sets.

Solution.

Using a Venn diagram one can easily see thatA\BandA\Bcare disjoint sets such thatA= (A\B)[(A\Bc)Example 2.7 Each team in a basketball league plays 20 games in one tournament. Event Ais the event that Team 1 wins 15 or more games in the tournament. Event Bis the event that Team 1 wins less than 10 games. EventCis the event that Team 1 wins between 8 to 16 games. Of course, Team 1 can win at most

20 games. Using words, what do the following events represent?

(a)A[BandA\B: (b)A[CandA\C: (c)B[CandB\C: (d)Ac; Bc;andCc:

Solution.

(a)A[Bis the event that Team 1 wins 15 or more games or wins 9 or less games.A\Bis the empty set, since Team 1 cannot win 15 or more games and have less than 10 wins at the same time. Therefore, eventAand event

Bare disjoint.

(b)A[Cis the event that Team 1 wins at least 8 games.A\Cis the event that Team 1 wins 15 or 16 games. (c)B[Cis the event that Team 1 wins at most 16 games.B\Cis the event that Team 1 wins 8 or 9 games. (d)Acis the event that Team 1 wins 14 or fewer games.Bcis the event that Team 1 wins 10 or more games.Ccis the event that Team 1 wins fewer than

8 or more than 16 games

2 SET OPERATIONS21

Given the setsA1;A2;;we dene

1n=1An=fxjx2Aifor all i2Ng:

Example 2.8

For each positive integernwe deneAn=fng:Find\1n=1An:

Solution.

Clearly,\1n=1An=;Remark 2.1

Note that the Venn diagrams ofA\BandA[Bshow thatA\B=B\A andA[B=B[A:That is,[and\are commutative laws. The following theorem establishes the distributive laws of sets.

Theorem 2.1

IfA;B;andCare subsets ofUthen

(a)A\(B[C) = (A\B)[(A\C): (b)A[(B\C) = (A[B)\(A[C):

Proof.

See Problem 2.16Remark 2.2

Note that since\and[are commutative operations then (A\B)[C= (A[C)\(B[C) and (A[B)\C= (A\C)[(B\C): The following theorem presents the relationships between (A[B)c;(A\

B)c;AcandBc:

Theorem 2.2(De Morgan's Laws)

LetAandBbe subsets ofUthen

(a) (A[B)c=Ac\Bc: (b) (A\B)c=Ac[Bc:

22BASIC OPERATIONS ON SETS

Proof.

We prove part (a) leaving part(b) as an exercise for the reader. (a) Letx2(A[B)c:Thenx2Uandx62A[B:Hence,x2Uand (x62A andx62B). This implies that (x2Uandx62A) and (x2Uandx62B).

It follows thatx2Ac\Bc:

Conversely, letx2Ac\Bc:Thenx2Acandx2Bc:Hence,x62Aandquotesdbs_dbs42.pdfusesText_42

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