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

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36 COUNTING AND COMBINATORICS Example 3.5 How many license-plates with 3 letters followed by 3 digits exist if exactly one of the digits is 1 Solution. In this case, we must pick a place for the 1 digit, and then the remaining digit places must be populated from the digits {0, 2, · · · 9}. A 6-step process: (1) Choose the first letter, (2) choose the second letter, (3) choose the third letter, (4) choose which of three positions the 1 goes, (5) choose the first of the other digits, and (6) choose the second of the other digits. Every step can be done in a number of ways that does not depend on previous choices, and each license plate can be specified in this manner. So there are 26 · 26 · 26 · 3 · 9 · 9 = 4, 270, 968 ways
3 THE FUNDAMENTAL PRINCIPLE OF COUNTING 37 Problems Problem 3.1 If each of the 10 digits is chosen at random, how many ways can you choose the following numbers (a) A two-digit code number, repeated digits permitted. (b) A three-digit identification card number, for which the first digit cannot be a 0. (c) A four-digit bicycle lock number, where no digit can be used twice. (d) A five-digit zip code number, with the first digit not zero. Problem 3.2 (a) If eight horses are entered in a race and three finishing places are considered, how many finishing orders can they finish Assume no ties. (b) If the top three horses are Lucky one, Lucky Two, and Lucky Three, in how many possible orders can they finish Problem 3.3 You are taking 3 shirts(red, blue, yellow) and 2 pairs of pants (tan, gray) on a trip. How many different choices of outfits do you have Problem 3.4 A club has 10 members. In how many ways can the club choose a president and vice-president if everyone is eligible Problem 3.5 In a medical study, patients are classified according to whether they have blood type A, B, AB, or O, and also according to whether their blood pressure is low (L), normal (N), or high (H). Use a tree diagram to represent the various outcomes that can occur. Problem 3.6 If a travel agency offers special weekend trips to 12 different cities, by air, rail, or bus, in how many different ways can such a trip be arranged Problem 3.7 If twenty paintings are entered in an art show, in how many different ways can the judges award a first prize and a second prize
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 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: 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
Page 69 and 70: 7 PROPERTIES OF PROBABILITY 69 Sinc
Page 71 and 72: 7 PROPERTIES OF PROBABILITY 71 Proo
Page 73 and 74: 7 PROPERTIES OF PROBABILITY 73 Prob
Page 75 and 76: 8 PROBABILITY AND COUNTING TECHNIQU
Page 81 and 82: Conditional Probability and Indepen
Page 83 and 84: 9 CONDITIONAL PROBABILITIES 83 Proo
Page 85 and 86: 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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