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

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48 COUNTING AND COMBINATORICS Problem 4.16 How many different 12-person juries can be chosen from a pool of 20 jurors Problem 4.17 A jeweller has 15 different sized pearls to string on a circular band. In how many ways can this be done Problem 4.18 Four teachers and four students are seated in a circular discussion group. Find the number of ways this can be done if teachers and students must be seated alternately.
5 PERMUTATIONS AND COMBINATIONS WITH INDISTINGUISHABLE OBJECTS49 5 Permutations and Combinations with Indistinguishable Objects So far we have looked at permutations and combinations of objects where each object was distinct (distinguishable). Now we will consider allowing multiple copies of the same (identical/indistinguishable) item. Permutations with Repetitions(i.e. Indistinguishable Objects) We have seen how to calculate basic permutation problems where each element can be used one time at most. In the following discussion, we will see what happens when elements can be used repeatedly. As an example, consider the following problem: How many different letter arrangements can be formed using the letters in DECEIVED Note that the letters in DECEIVED are not all distinguishable since it contains repeated letters such as E and D. Thus, interchanging the second and the fourth letters gives a result indistinguishable from interchanging the second and the seventh letters. The number of different rearrangement of the letters in DECEIVED can be found by applying the following theorem. Theorem 5.1 Given n objects of which n 1 indistinguishable objects of type 1, n 2 indistinguishable objects of type 2, · · · , n k indistinguishable objects of type k where n 1 + n 2 + · · · + n k = n. Then the number of distinguishable permutations of n objects is given by: n! n 1 !n 2 ! · · · n k ! . Proof. The task of forming a permutation can be divided into k subtasks: For the first kind we have n slots and n 1 objects to place such that the order is unimportant. There are C(n, n 1 ) different ways. For the second kind we have n − n 1 slots and n 2 objects to place such that the order is unimportant. There are C(n−n 1 , n 2 ) ways. For the k th kind we have n−n 1 −n 2 −· · ·−n k−1 slots and n k objects to place such that the order is unimportant. There are C(n − n 1 − n 2 − · · · − n k−1 , n k ) different ways. Thus, applying the Fundamental Principle of Counting we find that the number of distinguishable permutations is C(n, n 1 ) · C(n − n 1 , n 2 ) · · · C(n − n 1 − · · · n k−1 , n k ) = n! n 1 !n 2 ! · · · n k !
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 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 47: 4 PERMUTATIONS AND COMBINATIONS 47
Page 51 and 52: 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
Page 87 and 88: 9 CONDITIONAL PROBABILITIES 87 Prob
Page 89 and 90: 10 POSTERIOR PROBABILITIES: BAYES
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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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