A Probability Course for the Actuaries A Preparation for Exam P/1

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18 BASIC OPERATIONS ON SETS Problem 1.8 Use induction to show that (1 + x) n ≥ 1 + nx for all n ≥ 1, where x > −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) first 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). Let S be the collection of all outcomes. List the elements of S and then find the cardinality of S.
2 SET OPERATIONS 19 2 Set Operations In this section we introduce various operations on sets and study the properties 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 A c = {x ∈ U|x ∉ A}. Example 2.1 Find the complement of A = {1, 2, 3} if U = {1, 2, 3, 4, 5, 6}. Solution. From the definition, A c = {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 ∉ 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. That is, A − B = {1, 2} Union and Intersection Given two sets A and B. The union of A and B is the set A ∪ B = {x|x ∈ A or x ∈ B}
Page 1 and 2: A Probability Course for the Actuar
Page 3 and 4: Contents Preface 7 Risk Management
Page 5 and 6: CONTENTS 5 40 The Central Limit The
Page 7 and 8: Preface The present manuscript is d
Page 9 and 10: RISK MANAGEMENT CONCEPTS 9 Two vari
Page 11 and 12: Basic Operations on Sets The axioma
Page 13 and 14: 1 BASIC DEFINITIONS 13 Example 1.3
Page 15 and 16: 1 BASIC DEFINITIONS 15 Let A and B
Page 17: 1 BASIC DEFINITIONS 17 Problems Pro
Page 21 and 22: 2 SET OPERATIONS 21 (g) A ∪ (A c
Page 23 and 24: 2 SET OPERATIONS 23 Given the sets
Page 25 and 26: 2 SET OPERATIONS 25 Example 2.11 Fi
Page 27 and 28: 2 SET OPERATIONS 27 Proof. Suppose
Page 29 and 30: 2 SET OPERATIONS 29 Of these policy
Page 31 and 32: 2 SET OPERATIONS 31 Problem 2.15
Page 33 and 34: Counting and Combinatorics The majo
Page 35 and 36: 3 THE FUNDAMENTAL PRINCIPLE OF COUN
Page 37 and 38: 3 THE FUNDAMENTAL PRINCIPLE OF COUN
Page 39 and 40: 4 PERMUTATIONS AND COMBINATIONS 39
Page 49 and 50: 5 PERMUTATIONS AND COMBINATIONS WIT
Page 59 and 60: Probability: Definitions and Proper
Page 61 and 62: 6 BASIC DEFINITIONS AND AXIOMS OF P
Page 67 and 68: 7 PROPERTIES OF PROBABILITY 67 7 Pr
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7 PROPERTIES OF PROBABILITY 69 Sinc
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7 PROPERTIES OF PROBABILITY 71 Proo
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7 PROPERTIES OF PROBABILITY 73 Prob
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8 PROBABILITY AND COUNTING TECHNIQU
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Conditional Probability and Indepen
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9 CONDITIONAL PROBABILITIES 83 Proo
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9 CONDITIONAL PROBABILITIES 85 Prob
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9 CONDITIONAL PROBABILITIES 87 Prob
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10 POSTERIOR PROBABILITIES: BAYES
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11 INDEPENDENT EVENTS 101 Example 1
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11 INDEPENDENT EVENTS 103 then by T
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11 INDEPENDENT EVENTS 105 Solution.
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11 INDEPENDENT EVENTS 107 independe
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12 ODDS AND CONDITIONAL PROBABILITY
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12 ODDS AND CONDITIONAL PROBABILITY
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Discrete Random Variables This chap
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13 RANDOM VARIABLES 115 Example 13.
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13 RANDOM VARIABLES 117 Problem 13.
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14 PROBABILITY MASS FUNCTION AND CU
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15 EXPECTED VALUE OF A DISCRETE RAN
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16 EXPECTED VALUE OF A FUNCTION OF
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17 VARIANCE AND STANDARD DEVIATION
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18 BINOMIAL AND MULTINOMIAL RANDOM
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19 POISSON RANDOM VARIABLE 161 (b)
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19 POISSON RANDOM VARIABLE 163 To f
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19 POISSON RANDOM VARIABLE 165 Usin
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19 POISSON RANDOM VARIABLE 167 Prob
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19 POISSON RANDOM VARIABLE 169 Prob
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20 OTHER DISCRETE RANDOM VARIABLES
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21 PROPERTIES OF THE CUMULATIVE DIS
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Continuous Random Variables Continu
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22 DISTRIBUTION FUNCTIONS 207 Solut
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22 DISTRIBUTION FUNCTIONS 209 Proof
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22 DISTRIBUTION FUNCTIONS 211 rando
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22 DISTRIBUTION FUNCTIONS 213 Probl
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22 DISTRIBUTION FUNCTIONS 215 Probl
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23 EXPECTATION, VARIANCE AND STANDA
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24 THE UNIFORM DISTRIBUTION FUNCTIO
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25 NORMAL RANDOM VARIABLES 241 Towa
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25 NORMAL RANDOM VARIABLES 243 Figu
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25 NORMAL RANDOM VARIABLES 245 (b)
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25 NORMAL RANDOM VARIABLES 247 tria
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25 NORMAL RANDOM VARIABLES 249 rand
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25 NORMAL RANDOM VARIABLES 251 Prob
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25 NORMAL RANDOM VARIABLES 253 Prob
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26 EXPONENTIAL RANDOM VARIABLES 255
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27 GAMMA AND BETA DISTRIBUTIONS 265
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28 THE DISTRIBUTION OF A FUNCTION O
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Joint Distributions There are many
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29 JOINTLY DISTRIBUTED RANDOM VARIA
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30 INDEPENDENT RANDOM VARIABLES 299
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31 SUM OF TWO INDEPENDENT RANDOM VA
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32 CONDITIONAL DISTRIBUTIONS: DISCR
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33 CONDITIONAL DISTRIBUTIONS: CONTI
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34 JOINT PROBABILITY DISTRIBUTIONS
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Properties of Expectation We have s
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36 COVARIANCE, VARIANCE OF SUMS, AN
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37 CONDITIONAL EXPECTATION 371 (b)
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37 CONDITIONAL EXPECTATION 373 If x
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37 CONDITIONAL EXPECTATION 375 in t
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37 CONDITIONAL EXPECTATION 377 Prob
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37 CONDITIONAL EXPECTATION 379 stat
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38 MOMENT GENERATING FUNCTIONS 381
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Limit Theorems Limit theorems are c
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39 THE LAW OF LARGE NUMBERS 399 Pro
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39 THE LAW OF LARGE NUMBERS 401 Sol
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39 THE LAW OF LARGE NUMBERS 403 Exa
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39 THE LAW OF LARGE NUMBERS 405 It
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39 THE LAW OF LARGE NUMBERS 407 Not
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39 THE LAW OF LARGE NUMBERS 409 We
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40 THE CENTRAL LIMIT THEOREM 415 No
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40 THE CENTRAL LIMIT THEOREM 417 So
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40 THE CENTRAL LIMIT THEOREM 419 X
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40 THE CENTRAL LIMIT THEOREM 421 Pr
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40 THE CENTRAL LIMIT THEOREM 423 pr
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41 MORE USEFUL PROBABILISTIC INEQUA
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Appendix 42 Improper Integrals A ve
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42 IMPROPER INTEGRALS 433 Figure 42
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42 IMPROPER INTEGRALS 435 Example 4
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42 IMPROPER INTEGRALS 437 • Impro
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43 DOUBLE INTEGRALS 439 We call L =
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43 DOUBLE INTEGRALS 441 Example 43.
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43 DOUBLE INTEGRALS 443 and, substi
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43 DOUBLE INTEGRALS 445 This means,
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43 DOUBLE INTEGRALS 447 Example 43.
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43 DOUBLE INTEGRALS 449 Problem 43.
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44 DOUBLE INTEGRALS IN POLAR COORDI
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Bibliography [1] R. Sheldon , A fir
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A Probability Course for the Actuaries A Preparation for Exam P/1

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