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Exam MAS I Study Manual
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TO OUR READERS, Please check A S M s web site at www studymanuals com for errata. and updates If you have any comments or reports of errata please. e mail us at mail studymanuals com, Copyright 2019 by Actuarial Study Materials A S M PO Box 69. Greenland NH 03840 All rights reserved Reproduction in whole or in part. without express written permission from the publisher is strictly prohibited. I Probability Review 1, 1 Probability Review 3, 1 1 Functions and moments 3. 1 2 Probability distributions 5, 1 2 1 Bernoulli distribution 5. 1 2 2 Uniform distribution 5, 1 2 3 Exponential distribution 5.
1 3 Variance 6, 1 4 Normal approximation 7, 1 5 Conditional probability and expectation 8. Exercises 11, Solutions 13, 2 Parametric Distributions 17. 2 1 Transformations 17, 2 2 Common parametric distributions 18. 2 2 1 Uniform 18, 2 2 2 Beta 19, 2 2 3 Exponential 20. 2 2 4 Weibull 20, 2 2 5 Gamma 21, 2 2 6 Pareto 22, 2 2 7 Single parameter Pareto 24.
2 2 8 Lognormal 24, Exercises 24, Solutions 26, 3 Mixtures 27. 3 1 Discrete mixtures 27, 3 2 Continuous mixtures 29. 3 3 Conditional variance 30, Exercises 32, Solutions 37. II Stochastic Processes 43, 4 Markov Chains Chapman Kolmogorov Equations 45. 4 1 Chapman Kolmogorov equations 46, 4 2 Gambler s ruin 49.
4 3 Algorithmic efficiency 50, Exercises 51, Solutions 59. 5 Markov Chains Classification of States 67, 5 1 Random walks 68. Exam MAS I Study Manual 1st edition 4th printing iii. Copyright 2019 ASM, iv CONTENTS, Exercises 68, Solutions 71. 6 Discrete Markov Chains Long Run Proportions and Limiting Probabilities 73. Exercises 75, Solutions 77, 7 Markov Chains Time in Transient States 83. Appendix Inverting a Matrix 86, Exercises 88, Solutions 92.
8 Markov Chains Branching Processes 95, Exercises 96. Solutions 98, 9 Markov Chains Time Reversible 101, Exercises 103. Solutions 104, 10 Exponential Distribution 107, 10 1 Lack of memory 107. 10 2 Minimum 109, 10 3 Sum 109, Exercises 110, Solutions 111. 11 The Poisson Process Probabilities of Events 113. 11 1 Introduction 113, 11 2 Probabilities homogeneous process 114.
11 3 Probabilities non homogeneous process 115, Exercises 118. Solutions 122, 12 The Poisson Process Time To Next Event 127. Exercises 129, Solutions 132, 13 The Poisson Process Thinning 135. 13 1 Constant probabilities 135, 13 2 Non constant probabilities 137. Exercises 138, Solutions 144, 14 The Poisson Process Other Characteristics 151.
14 1 Probability of events from one system before events from another system 151. 14 2 Conditional distribution of event time 151, Exercises 152. Solutions 153, 15 The Poisson Process Sums and Mixtures 157. 15 1 Sums of Poisson processes 157, 15 2 Mixtures of Poisson processes 158. Exercises 162, Exam MAS I Study Manual 1st edition 4th printing. Copyright 2019 ASM, CONTENTS v, Solutions 165, 16 Compound Poisson Processes 171.
16 1 Definition and moments 171, 16 2 Sums of compound distributions 173. Exercises 174, Solutions 180, 17 Reliability Structure Functions 187. 17 1 Expressing a system as a parallel set of series structures 188. 17 2 Expressing a system as a series set of parallel structures 190. Exercises 191, Solutions 192, 18 Reliability Probabilities 195. 18 1 Exact calculation of the reliability function 195. 18 2 Upper and lower bounds for the reliability function 196. 18 2 1 Inclusion Exclusion Bounds 196, 18 2 2 Estimating Intersection Probabilities 200. Exercises 201, Solutions 203, 19 Reliability Time to Failure 207.
19 1 Expected amount of time to failure 207, 19 2 Increasing failure rates 208. Exercises 209, Solutions 211, III Life Contingencies 213. 20 Survival Models 215, 20 1 Survival probabilities 215. 20 2 Special situations 217, 20 3 Life expectancy 218. Exercises 219, Solutions 222, 21 Contingent Payments 225.
21 1 Life insurances 225, 21 2 Life annuities 227, 21 3 Premiums 229. Exercises 230, Solutions 239, IV Simulation 247, 22 Simulation Inverse Transformation Method 249. Exercises 251, Solutions 257, 23 Simulation Applications 265. Exercises 266, Exam MAS I Study Manual 1st edition 4th printing. Copyright 2019 ASM, vi CONTENTS, Solutions 270, 24 Simulation Rejection Method 275.
Exercises 278, Solutions 280, V Statistics 285, 25 Estimator Quality 287. 25 1 Bias 287, 25 2 Consistency 288, 25 3 Efficiency and mean square error 289. Exercises 290, Solutions 297, 26 Kernel Density Estimation 303. 26 1 Empirical distribution 303, 26 2 Kernel functions 303. 26 2 1 Rectangular kernel 304, 26 2 2 Triangular kernel 309.
26 2 3 Gaussian kernel 315, 26 2 4 Other kernels 317. 26 3 Moments of kernel smoothed distributions 318, Exercises 319. Solutions 326, 27 Method of Moments 335, 27 1 Introduction to parametric estimation 335. 27 2 The method of moments 336, 27 2 1 Exponential 336. 27 2 2 Gamma 336, 27 2 3 Pareto 337, 27 2 4 Lognormal 338.
27 2 5 Uniform 339, 27 2 6 Other distributions 339. 27 3 Fitting other moments 340, 27 4 Fitting incomplete data 340. Exercises 342, Solutions 350, 28 Percentile Matching 363. 28 1 Smoothed empirical percentile 363, 28 2 Percentile matching for various distributions 363. 28 2 1 Exponential 364, 28 2 2 Weibull 364, 28 2 3 Lognormal 365.
28 2 4 Other distributions 366, 28 3 Percentile matching with incomplete data 366. 28 4 Matching a percentile and a moment 368, Exercises 368. Solutions 376, 29 Maximum Likelihood Estimators 387. Exam MAS I Study Manual 1st edition 4th printing, Copyright 2019 ASM. CONTENTS vii, 29 1 Defining the likelihood 388, 29 1 1 Individual data 388.
29 1 2 Grouped data 389, 29 1 3 Censoring 390, 29 1 4 Truncation 391. 29 1 5 Combination of censoring and truncation 392. Exercises 392, Solutions 403, 30 Maximum Likelihood Estimators Special Techniques 413. 30 1 Cases for which the Maximum Likelihood Estimator equals the Method of Moments Estimator 413. 30 1 1 Exponential distribution 413, 30 2 Parametrization and Shifting 414. 30 2 1 Parametrization 414, 30 2 2 Shifting 415, 30 3 Transformations 415. 30 3 1 Lognormal distribution 415, 30 3 2 Inverse exponential distribution 416.
30 3 3 Weibull distribution 417, 30 4 Special distributions 417. 30 4 1 Uniform distribution 417, 30 4 2 Pareto distribution 418. 30 4 3 Beta distribution 420, 30 5 Bernoulli technique 420. Exercises 424, Solutions 439, 31 Variance of Maximum Likelihood Estimator 455. Exercises 456, Solutions 457, 32 Sufficient Statistics 459.
Exercises 461, Solutions 463, 33 Hypothesis Testing 465. 33 1 Background 465, 33 2 Typical exam questions 468. 33 2 1 Calculate significance or power 468, 33 2 2 Determine critical values 470. Exercises 472, Solutions 480, 34 Confidence Intervals and Sample Size 487. 34 1 Confidence intervals 487, 34 2 Sample size 488.
Exercises 489, Solutions 491, 35 Confidence Intervals for Means 495. 35 1 2 distribution 495, 35 2 Student s t distribution 496. 35 3 Testing the mean of a Bernoulli population 497. 35 4 Testing the difference of means from two populations 498. Exam MAS I Study Manual 1st edition 4th printing, Copyright 2019 ASM. viii CONTENTS, 35 4 1 Two unpaired normal populations 498. 35 4 2 Two paired normal populations 499, 35 4 3 Two Bernoulli populations 500.
Exercises 502, Solutions 508, 36 Kolmogorov Smirnov Tests 513. Exercises 517, Solutions 522, 37 Chi Square Tests 529. 37 1 One dimensional chi square 529, 37 2 Two dimensional chi square 532. Exercises 533, Solutions 539, 38 Confidence Intervals for Variances 545. 38 1 Testing variances 545, 38 2 Testing ratios of variances the F distribution 546.
Exercises 548, Solutions 552, 39 Uniformly Most Powerful Critical Regions 555. Exercises 557, Solutions 559, 40 Likelihood Ratio Tests 563. Exercises 565, Solutions 569, 41 Order Statistics 573. Exercises 577, Solutions 579, 42 q q Plots 585, Exercises 586. Solutions 587, VI Extended Linear Model 589, 43 Introduction to Extended Linear Models 591.
43 1 Statistical learning 591, 43 2 Types of variables 591. 43 3 Graphs 592, Exercises 593, Solutions 595, 44 How a Generalized Linear Model Works 597. 44 1 Basic models 597, 44 2 Exponential family 599. 44 3 Poisson response 603, Exercises 603, Solutions 608. Exam MAS I Study Manual 1st edition 4th printing, Copyright 2019 ASM.
CONTENTS ix, 45 How a Generalized Linear Model Works Categorical Response 611. 45 1 Binomial response 611, 45 2 Nominal Response 615. 45 3 Ordinal response 616, 45 4 An alternative to logistic regression 619. 45 5 Classification 621, Exercises 624, Solutions 633. 46 Generalized Linear Model Estimating Parameters 641. Exercises 647, Solutions 653, 47 Generalized Linear Model Measures of Fit 661.
47 1 Deviance 661, 47 2 Three tests for evaluating significance of explanatory variables 664. 47 3 Penalized loglikelihood tests 667, Exercises 668. Solutions 673, 48 Normal Linear Model Standard Error R 2 and t statistic 677. 48 1 Standard error of the regression 677, 48 2 R 2 the coefficient of determination 679. 48 3 t statistic 680, Exercises 681, Solutions 689.
49 Normal Linear Model F and VIF 695, 49 1 F statistic 695. 49 2 Collinearity of explanatory variables VIF 698. Exercises 699, Solutions 709, 50 Normal Linear Model Validation 715. 50 1 Validating model assumptions 715, 50 2 Influential points 717. Exercises 718, Solutions 721, 51 Normal Linear Model Predictions 725. Exercises 725, Solutions 726, 52 ANOVA 729, 52 1 One factor ANOVA 729.
52 2 Two factor ANOVA without replication 734, 52 3 Two factor ANOVA with replication 736. 52 4 ANCOVA 740, 52 5 Log linear models 740, Exercises 741. Solutions 755, 53 Generalized Linear Models Measures of Fit II 765. 53 1 Goodness of fit tests 765, Exam MAS I Study Manual 1st edition 4th printing. Copyright 2019 ASM, x CONTENTS, 53 2 Residuals 768.
53 3 Overdispersion 769, Exercises 769, Solutions 774. 54 Resampling Methods 779, 54 1 Cross validation 780. 54 1 1 Validation set approach 780, 54 1 2 Cross validation 780. 54 2 The Bootstrap 781, Exercises 782, Solutions 785. 55 Normal Linear Model Subset Selection 787, 55 1 Subset selection 787.
55 2 Choosing the best model 788, Exercises 789, Solutions 794. 56 Normal Linear Model Shrinkage and Dimension Reduction 799. 56 1 Shrinkage methods 799, 56 1 1 Ridge regression 799. 56 1 2 The lasso 800, 56 2 Dimension reduction methods 802. 56 2 1 Principal components regression 802, 56 2 2 Partial least squares 802. 56 3 The curse of dimensionality 803, Exercises 803.
Solutions 807, 57 Extensions to the Linear Model 811. 57 1 Extensions involving linear expressions with functions of explanatory variables 811. 57 2 Splines 811, 57 2 1 Regression splines 811, 57 2 2 Smoothing splines 812. 57 3 Local regression 813, 57 4 Generalized additive models 813. Exercises 814, Solutions 818, VII Time Series 819, 58 Time Series Trend and Seasonality 821. Exercises 824, Solutions 825, 59 Time Series Correlation 827.
59 1 Second order properties of a time series 827, 59 2 Relationships of different time series 829. Exercises 831, Solutions 833, 60 Time Series White Noise and Random Walks 837. Exam MAS I Study Manual 1st edition 4th printing, Copyright 2019 ASM. CONTENTS xi, 60 1 White noise 837, 60 2 Random walks 837. Exercises 838, Solutions 839, 61 Time Series Autoregressive Models 841.
Exercises 845, Solutions 848, 62 Time Series Regression 851. 62 1 Correcting for autocorrelation 851, 62 2 Seasonality 852. 62 3 Logarithmic transformations 853, 62 4 Error correction factors 853. Exercises 854, Solutions 857, 63 Time Series Moving Average Models 861. Exercises 864, Solutions 868, 64 Time Series ARMA Models 873.
Exercises 875, Solutions 880, 65 Time Series ARIMA and SARIMA models 885. Exercises 886, Solutions 888, VIII Practice Exams 891. 1 Practice Exam 1 893, 2 Practice Exam 2 907, 3 Practice Exam 3 921. 4 Practice Exam 4 935, 5 Practice Exam 5 951, 6 Practice Exam 6 967. Appendices 983, A Solutions to the Practice Exams 985.
Solutions for Practice Exam 1 985, Solutions for Practice Exam 2 994. Solutions for Practice Exam 3 1004, Solutions for Practice Exam 4 1013. Solutions for Practice Exam 5 1023, Solutions for Practice Exam 6 1035. Exam MAS I Study Manual 1st edition 4th printing, Copyright 2019 ASM. xii CONTENTS, B Solutions to Statistics and Stochastic Process Questions on Old CAS 3 and 3L Exams 1045.
B 1 Solutions to CAS Exam 3 Spring 2005 1045, B 2 Solutions to CAS Exam 3 Fall 2005 1049. B 3 Solutions to CAS Exam 3 Spring 2006 1051, B 4 Solutions to CAS Exam 3 Fall 2006 1054. B 5 Solutions to CAS Exam 3 Spring 2007 1057, B 6 Solutions to CAS Exam 3 Fall 2007 1059. B 7 Solutions to CAS Exam 3L Spring 2008 1061, B 8 Solutions to CAS Exam 3L Fall 2008 1063. B 9 Solutions to CAS Exam 3L Spring 2009 1066, B 10 Solutions to CAS Exam 3L Fall 2009 1069.
Exam MAS I Study Manual This manual includes Customizable versatile online exam question bank Thousands of questions Access your exclusive StudyPlus bonus content GOAL Flashcards Career Guides Key Code Inside TO OUR READERS Please check A S M s web site at www studymanuals com for errata and updates If you have any comments or reports of errata please e mail us at mail

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