Lecture Notes in Actuarial Mathematics A Probability Course for the Actuaries A Preparation for

765 Pages·2017·3.29 MB·English

Lecture Notes in Actuarial
Mathematics
A Probability Course for the
Actuaries
A Preparation for Exam P/1
Marcel B. Finan
May 2018 Syllabus
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 (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: mfi[email protected]
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, 2017
i
ii PREFACE
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
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 Variable345
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
CONTENTS 3
45 Joint Probability Distributions of Functions of Random Variables435
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
4 CONTENTS
Set Theory Prerequisite
Two approaches of the concept of probability will be introduced later in the
book: Theclassicalprobabilityandtheexperimentalprobability. Theformer
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 (cid:54)= 0}.
b
• The set R of all real numbers.
5
6 SET THEORY PREREQUISITE
1 Some Basic Definitions
We define a set as a collection of well-defined objects (called elements or
membersofA)suchthatforanygivenobjectonecanassertwithoutdispute
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 (cid:54)∈ 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