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

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22 BASIC OPERATIONS ON SETS Solution. Using a Venn diagram one can easily see that [(A∪B)∩(A∪C)∩(B c ∩C c )] = A − [A ∩ (B ∪ C)] If A ∩ B = ∅ we say that A and B are disjoint sets. Example 2.6 Let A and B be two non-empty sets. Write A as the union of two disjoint sets. Solution. Using a Venn diagram one can easily see that A ∩ B and A ∩ B c are disjoint sets such that A = (A ∩ B) ∪ (A ∩ B c ) Example 2.7 Each team in a basketball league plays 20 games in one tournament. Event A is the event that Team 1 wins 15 or more games in the tournament. Event B is the event that Team 1 wins less than 10 games. Event C is 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 ∪ B and A ∩ B. (b) A ∪ C and A ∩ C. (c) B ∪ C and B ∩ C. (d) A c , B c , and C c . Solution. (a) A ∪ B is the event that Team 1 wins 15 or more games or wins 9 or less games. A ∩ B is the empty set, since Team 1 cannot win 15 or more games and have less than 10 wins at the same time. Therefore, event A and event B are disjoint. (b) A ∪ C is the event that Team 1 wins at least 8 games. A ∩ C is the event that Team 1 wins 15 or 16 games. (c) B ∪ C is the event that Team 1 wins at most 16 games. B ∩ C is the event that Team 1 wins 8 or 9 games. (d) A c is the event that Team 1 wins 14 or fewer games. B c is the event that Team 1 wins 10 or more games. C c is the event that Team 1 wins fewer than 8 or more than 16 games
2 SET OPERATIONS 23 Given the sets A 1 , A 2 , · · · , we define ∩ ∞ n=1A n = {x|x ∈ A i for all i ∈ N}. Example 2.8 For each positive integer n we define A n = {n}. Find ∩ ∞ n=1A n . Solution. Clealry, ∩ ∞ n=1A n = ∅ Remark 2.1 Note that the Venn diagrams of A ∩ B and A ∪ B show that A ∩ B = B ∩ A and A ∪ B = B ∪ A. That is, ∪ and ∩ are commutative laws. The following theorem establishes the distributive laws of sets. Theorem 2.1 If A, B, and C are subsets of U then (a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). (b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C). Proof. See Problem 2.16 Remark 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 , A c and B c . Theorem 2.2 (De Morgan’s Laws) Let A and B be subsets of U then (a) (A ∪ B) c = A c ∩ B c . (b) (A ∩ B) c = A c ∪ B c .
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 and 18: 1 BASIC DEFINITIONS 17 Problems Pro
Page 19 and 20: 2 SET OPERATIONS 19 2 Set Operation
Page 21: 2 SET OPERATIONS 21 (g) A ∪ (A c
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
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Page 69 and 70: 7 PROPERTIES OF PROBABILITY 69 Sinc
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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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