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

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4 CONTENTS 17 Variance and Standard Deviation . . . . . . . . . . . . . . . . . 140 18 Binomial and Multinomial Random Variables . . . . . . . . . . . 146 19 Poisson Random Variable . . . . . . . . . . . . . . . . . . . . . . 160 20 Other Discrete Random Variables . . . . . . . . . . . . . . . . . 170 20.1 Geometric Random Variable . . . . . . . . . . . . . . . . 170 20.2 Negative Binomial Random Variable . . . . . . . . . . . . 177 20.3 Hypergeometric Random Variable . . . . . . . . . . . . . 184 21 Properties of the Cumulative Distribution Function . . . . . . . 190 Continuous Random Variables 205 22 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . 205 23 Expectation, Variance and Standard Deviation . . . . . . . . . . 217 24 The Uniform Distribution Function . . . . . . . . . . . . . . . . 235 25 Normal Random Variables . . . . . . . . . . . . . . . . . . . . . 240 26 Exponential Random Variables . . . . . . . . . . . . . . . . . . . 255 27 Gamma and Beta Distributions . . . . . . . . . . . . . . . . . . 265 28 The Distribution of a Function of a Random Variable . . . . . . 277 Joint Distributions 285 29 Jointly Distributed Random Variables . . . . . . . . . . . . . . . 285 30 Independent Random Variables . . . . . . . . . . . . . . . . . . 299 31 Sum of Two Independent Random Variables . . . . . . . . . . . 310 31.1 Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . 310 31.2 Continuous Case . . . . . . . . . . . . . . . . . . . . . . . 315 32 Conditional Distributions: Discrete Case . . . . . . . . . . . . . 324 33 Conditional Distributions: Continuous Case . . . . . . . . . . . . 331 34 Joint Probability Distributions of Functions of Random Variables 340 Properties of Expectation 347 35 Expected Value of a Function of Two Random Variables . . . . . 347 36 Covariance, Variance of Sums, and Correlations . . . . . . . . . 357 37 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . 370 38 Moment Generating Functions . . . . . . . . . . . . . . . . . . . 381 Limit Theorems 397 39 The Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . 397 39.1 The Weak Law of Large Numbers . . . . . . . . . . . . . 397 39.2 The Strong Law of Large Numbers . . . . . . . . . . . . . 403
CONTENTS 5 40 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 414 41 More Useful Probabilistic Inequalities . . . . . . . . . . . . . . . 424 Appendix 431 42 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 431 43 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 44 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . 451 BIBLIOGRAPHY 456
Page 1 and 2: A Probability Course for the Actuar
Page 3: Contents Preface 7 Risk Management
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 49 and 50: 5 PERMUTATIONS AND COMBINATIONS WIT
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5 PERMUTATIONS AND COMBINATIONS WIT
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5 PERMUTATIONS AND COMBINATIONS WIT
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Probability: Definitions and Proper
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6 BASIC DEFINITIONS AND AXIOMS OF P
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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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