ACTEX Study Manual For CAS Exam MAS-I
ACTEXCAS Exam MAS-I Study ManualWith StudyPlus StudyPlus gives you digital access* to: Actuarial Exam & Career Guidebooks Technical Skill eLearning Tools And more!*See inside for keycode access and login instructionsSpring 2018 Edition Volume I-IIIAmbrose Lo, FSA, Ph.D., CERAACTEX Learning Learn Today. Lead Tomorrow.
ACTEX LearningLearn Today. Lead Tomorrow.Actuarial & Financial Risk Resource MaterialsSince 1972Copyright 2018 SRBooks, Inc.ISBN: 978-1-63588-230-8Printed in the United States of America.No portion of this ACTEX Study Manual may bereproduced or transmitted in any part or by any meanswithout the permission of the publisher.
PrefaceExam MAS-I (Modern Actuarial Statistics I ) is a new exam which will be offered for the firsttime in Spring 2018 on May 7, 2018 (Monday) by the Casualty Actuarial Society (CAS). Itreplaces its predecessor Exam S (Statistics and Probabilistic Models), which is a relativelyshort-lived exam offered only five times, from Fall 2015 to Fall 2017. Exam S, in turn,was developed from the old Exam LC (Models for Life Contingencies), Exam ST (Modelsfor Stochastic Processes and Statistics), and Exam 3L (Life Contingencies and Statistics).The introduction of Exam MAS-I is in response to the discontinuation of Exam C/4 inJuly 2018, which the CAS sees as an opportunity to revamp the current Exam S with aheavier focus on contemporary statistical methods and the addition of statistical learning.The construction of MAS-I is a further step that the CAS has taken to enhance actuaries’statistical literacy, as the CAS finds a deep understanding of statistics to be more importantfor property and casualty actuaries than routine training in traditional actuarial subjectslike life contingencies. You will considerably sharpen your statistics toolkit as a result oftaking (and, in all likelihood, passing!) Exam MAS-I.SyllabusThe syllabus of Exam MAS-I, available from .pdf, is extremely broad (but not necessarily deep) in scope, covering miscellaneous topics in applied probability, mathematical statistics, statistical modeling and timeseries analysis, many of which are new topics not tested in any SOA/CAS past exams. Asa rough estimate, you need at least three months of intensive study to master the materialin this exami . The specific sections of the syllabus along with their approximate weights inthe exam are shown below:SectioniRange of WeightA. Probability Models (Stochastic Processes & Survival Models)20–35%B. Statistics15–30%It is true that one need not master every topic in order to pass this exam.i
iiC. Extended Linear Models30–50%D. Time Series with Constant Variance10–20%Compared with the former Exam S, both Sections C and D have enjoyed a heavier weight,from 25–40% to 30–50% and from 5–10% to 10–20%, respectively. Sections A, B, and D aremore or less taken intact from the syllabuses of Exam S, ST, LC, and 3L. As a result, you canfind lots of relevant past exam questions on these two sections. Section C has experiencedthe biggest change, with a new textbook on statistical learning added.Exam FormatExam MAS-I is a four-hour multiple-choice exam. According to the list of MAS-I frequentlyasked questions (http://www.casact.org/cms/files/New CAS Exams MAS I and II FAQs1.pdf), the exam will consist of approximately 35 to 40 questions. Before the start of theexam, there will be a fifteen-minute reading period in which you can silently read the questions and check the exam booklet for missing or defective pages. However, writing will notbe permitted during this time, neither will the use of calculators.Given the similarity between Exam MAS-I and Exam S (whose syllabus is availablefrom f) in terms of their structure and topics, we may use Exam S as a rough proxy for Exam MAS-I. Each of the ExamS papers from Fall 2015 to Fall 2017 has 45 questions, categorized into the four sections asfollows:SectionA. Probability ModelsNumber of Questions2015F 2016S 2016F 2017S 2017F1312161516(Stochastic Processes & Survival Models)B. Statistics1715141314C. Generalized Linear Models1115111412D. Time Series with Constant Variance43433Total4545454545You can see that roughly the same number of exam questions was set on Sections A, B,and C, although Section C was proclaimed to be the most important section (perhaps eventhe examiners found it hard to set questions on this section!). To investigate whether sucha distribution of exam questions is consistent with the distribution that the CAS announcedin the exam syllabus, please try Practice Exam 2 Question 22 on page 1279. According .pdf, the pass marks for Fall 2015,Copyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
iiiSpring 2016, Fall 2016, and Spring 2017ii were 52.5, 55.0, 54.0, and 50.50 respectively,which means that candidates needed to answer about 27 out of 45 questions correctlyto earn a pass (each question carries 2 points with the total score being 44 2 88 or45 2 90).Here are the characteristics of a typical CAS multiple-choice exam:1. The questions are almost always arranged in the same order as the topics in the examsyllabus, so Question 1 is very likely a Poisson process question and Question 40 isa time series question. This implicitly gives you a hint as to which topic an examquestion is testing.2. A number of exam questions bear a striking resemblance to past CAS exam questions,sometimes even with the same numerical values. This attests to the importance ofpracticing numerous past exam problems, an abundance of which are discussed andsolved in this study manual in full detail.3. The scope of an exam can be narrow at times with several questions testing the sametopic in much the same way. For example, Questions 43, 44, and 45 of the Fall 2016Exam S all test time series forecasting for AR models, and Questions 21, 23, and 24 ofthe Fall 2017 Exam S all test the concepts of Type I and II errors.4. Most answer choices are in the form of ranges, e.g.:A. Less than 1%B. At least 1%, but less than 2%C. At least 2%, but less than 3%D. At least 3%, but less than 4%E. At least 4%If your answer is much lower than the bound indicated by Answer A or much higherthan that suggested by Answer E, do check your calculations. Chances are that youhave made computational mistakes, but this is not definitely the case (sometimes theCAS examiners themselves made a mistake!).Note that unlike other multiple-choice exams you took before, a guessing adjustmentwill be in place in Exam MAS-I, so unless you can eliminate two or three of the answerchoices, it will be wise of you not to answer questions which you are unsure of by pureguesswork.What is Special about This Study Manual?The objective of this study manual is to help you grasp the material in Exam MAS-I, whichis a brand new exam, effectively and efficiently, and pass it with considerable ease. Here aresome of the valuable features of this manual for achieving this all-important goal:iiThe pass mark for the Fall 2017 Exam S is not yet available when this manual goes in press.Copyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
iv Each chapter and section starts by explicitly stating which learning outcomes andknowledge statements of the exam syllabus we are going to cover, to ensure that weare on track and hitting the right target. The knowledge statements of the syllabus are demystified by precise and concise expositions synthesized from the syllabus readings, helping you acquire a deep understandingof the subject matter. Formulas and results of utmost importance are boxed for easy identification andmemorization. While the focus of this study manual is on exam preparation, it is interspersed withthe interpretation of various formulas that also fosters genuine understanding of thesyllabus material and discourages rote learning. It is the author’s belief and personalexperience that a solid understanding of the subject is always conducive to achievinggood exam results. Mnemonics and shortcuts are emphasized, so are highlights of important exam itemsand common mistakes committed by students. To succeed in any (actuarial) exam, the importance of practicing a wide variety of nontrivial problems to sharpen your understanding and to develop proficiency, as always,cannot be overemphasized. In this regard, there are no better learning resources thanpast Exam S and relevant SOA/CAS examination papers. The expositions of thismanual are complemented by more than 550 in-text examples and 750 end-ofchapter/section problems (the harder ones are labeled as [HARDER!] or [VERYHARD!!]), which are original or taken from relevant past exams and/or requiredtextbooks, all with step-by-step solutions and problem-solving remarks, to give youa sense of what you can expect to see in the real exam. All practice problems arephrased in a multiple-choice format to emulate the real exam environment. As youread this manual, skills are honed and confidence is built. As a general guide, youshould read through all of the in-text examples with particular attention paid to recentExam S questions, and work out at least half of the end-of-chapter/section problems. Three full-length practice exams updated for the MAS-I exam syllabus and designedto mimic the real exam conclude this study manual giving you a holistic review of thesyllabus material.New to the Spring 2018 Edition Old SOA/CAS exam questions before 2000 are added as appropriate. Despite their seniority, these old exam questions, which are not easily available nowadays, do illustratesome less commonly tested concepts and are of considerable value.Copyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
v All of the 45 questions from the very recent Fall 2017 Exam S are inserted into thismanual and carefully discussed and solved. Variants of some of these exam questionsare developed. A number of sections have been substantially revised, partly in response to the recentExam S papers, e.g. Section 1.2 on the properties of the exponential distribution,Section 10.6 on model diagnostics, Section 12.2 on the method of scoring. A numberof new examples and end-of-chapter/section problems have been added. In Bayesianparlance, we learn from experience as life moves on! All known typographical errors have been fixed.Exam TablesIn the exam, you will be supplied with a variety of tables, including: Standard normal distribution table (used throughout this study manual)You will need this table for values of the standard normal distribution function orstandard normal quantiles, when you work with normally distributed random variablesor perform normal approximation. Illustrative Life Table (used mostly in Chapter 4 of this study manual)You will need this when you are told that mortality of the underlying population followsthe Illustrative Life Table. A table of distributions for a number of common continuous and discrete distributionsand the formulas for their moments and other probabilistic quantities (used throughoutParts I and II of this study manual)This big table provides a great deal of information about some common as well asnon-common distributions (e.g. inverse exponential, inverse Gaussian, Pareto, Burr,etc.). When an exam question centers on these distributions and quantities such astheir means or variances are needed, consult this table. Quantiles of t-distribution, F -distribution iii , chi-square distribution (used in Chapters8, 10, 12 and 13 of this study manual)These quantiles will be of use when you perform parametric hypothesis tests.You should download these tables from http://www.casact.org/admissions/syllabus/MASI Tables.pdf right away, print out a copy and learn how to locate the relevant entries inthese tables because they will be intensively used during your study as well as in the exam.iiiThe F -table has a duplicate page.Copyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
viAcknowledgmentI would like to thank my colleagues, Professor Elias S. W. Shiu and Dr. Michelle A. Larson,at the University of Iowa for sharing with me many pre-2000 SOA/CAS exam papers. Thesehard-earned old exam papers have proved invaluable in illustrating a number of less commonly tested exam topics. Thanks are also due to Mr. Zhaofeng Tang, doctoral student inactuarial science at the University of Iowa, for his professional assistance in the productionof some of the graphs in this study manual.ErrataWhile we go to great lengths to polish and proofread this manual, some mistakes will inevitably go unnoticed. The author wishes to apologize in advance for any errors, typographical or otherwise, and would greatly appreciate it if you could bring them to hisattention by sending any errors you identify or any criticisms or compliments about thismanual to ambrose-lo@ uiowa. edu and c.c. support@ actexmadriver. com . The authorwill try his best to respond to any inquiries within 48 hours and an ongoing errata list willbe maintained online at ions/cas-exam-mas-I-study-manual. More importantly, students who report errors will qualityfor a quarterly drawing for a 100 in-store credit.Ambrose LoJanuary 2018Iowa City, IACopyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
viiAbout the AuthorProfessor Ambrose Lo was born, raised, and educated in Hong Kong. He earned his B.S. inActuarial Science (first class honors) and Ph.D. in Actuarial Science from The University ofHong Kong in 2010 and 2014 respectively. He joined the Department of Statistics and Actuarial Science at The University of Iowa in August 2014 as an Assistant Professor in ActuarialScience. He is a Fellow of the Society of Actuaries (FSA) and a Chartered Enterprise RiskAnalyst (CERA). His research interests lie in dependence structures, quantitative risk management as well as optimal (re)insurance. His research papers have been published in top-tieractuarial journals, such as Insurance: Mathematics and Economics, Scandinavian ActuarialJournal and ASTIN Bulletin: The Journal of the International Actuarial Association.Besides dedicating himself to actuarial research, Ambrose attaches equal importance toteaching, through which he nurtures the next generation of actuaries and serves the actuarialprofession. He has taught courses on financial derivatives, mathematical finance, life contingencies, credibility theory, advanced probability theory, and regression analysis. His emphasisin teaching is always placed on thorough understanding of the subject matter complementedby concrete problem-solving skills. As a result of his exceptional teaching performance, Ambrose has won numerous teaching awards ever since he was a graduate student (see e-teaching-assistant-2011-12-2 forexample).Copyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
viiiCopyright 2018 ACTEX LearningACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
ContentsPrefaceiI Probability Models (Stochastic Processes and SurvivalModels)11 Poisson Processes1.1 Fundamental Properties . . . . . . . .Problems . . . . . . . . . . . . . . . . .1.2 Hazard Rate Function . . . . . . . . .Problems . . . . . . . . . . . . . . . . .1.3 Inter-arrival and Waiting DistributionsProblems . . . . . . . . . . . . . . . . .1.4 New Poisson Processes from Old Ones1.4.1 Thinning . . . . . . . . . . . . .1.4.2 Sums of Poisson Processes . . .1.4.3 Problems . . . . . . . . . . . .1.5 Compound Poisson Processes . . . . .Problems . . . . . . . . . . . . . . . . .2 Reliability Theory2.1 Typical Systems . . . . . . . . . . . . . . . . . . . .2.2 Reliability of Systems of Independent Components .2.3 Expected System Lifetime . . . . . . . . . . . . . .2.4 End-of-chapter Problems . . . . . . . . . . . . . . .3 Markov Chains3.1 Transition Probabilities . . . . . . . . . . . . . . .3.1.1 Fundamentals . . . . . . . . . . . . . . . .3.1.2 Chapman–Kolmogorov Equations . . . . .3.1.3 [HARDER!] Path-dependent probabilities3.1.4 Problems . . . . . . . . . . . . . . . . . .3.2 The Gambler’s Ruin Problem . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . 03204204212217222260269
xCONTENTS3.33.43.53.63.7Classification of States . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . .Long-Run Proportions and Limiting ProbabilitiesProblems . . . . . . . . . . . . . . . . . . . . . . .Mean Time Spent in Transient States . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . .Branching Processes . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . .Time Reversible Markov Chains . . . . . . . . . .4 Elementary Life Contingencies4.1 Life Tables . . . . . . . . . . .4.2 Life Insurances . . . . . . . .4.3 Life Annuities . . . . . . . . .4.4 Net Level Premiums . . . . .4.5 End-of-chapter Problems . . .274282285299328334344349351.3553563633713813865 Simulation3995.1 Inversion Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4005.2 Acceptance-Rejection Method . . . . . . . . . . . . . . . . . . . . . . . . . . 4055.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409IIStatistics4196 Claim Severity Distributions4236.1 Continuous Distributions for Modeling Claim Severity . . . . . . . . . . . . . 4236.2 Mixture Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4276.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4297 Parametric Estimation7.1 Quality of Estimators . . . . . . . . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . .7.2 Some Parametric Estimation Methods . . . . . . . . .7.2.1 Method of Moments . . . . . . . . . . . . . . .7.2.2 Method of Percentile Matching . . . . . . . . .7.2.3 Problems . . . . . . . . . . . . . . . . . . . . .7.3 Maximum Likelihood Estimation . . . . . . . . . . . .7.3.1 Determining MLEs by Calculus . . . . . . . . .7.3.2 Distributions Which Require Special Techniques7.3.3 Remarks on MLE . . . . . . . . . . . . . . . . .7.3.4 Problems . . . . . . . . . . . . . . . . . . . . .7.4 Variance of Maximum Likelihood Estimators . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . .Copyright 2018 ACTEX 5ACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
CONTENTS7.5xiSufficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5858 Hypothesis Testing8.1 Fundamental Ideas of Hypothesis Testing . .Problems . . . . . . . . . . . . . . . . . . . .8.2 Test for Means of Normal Populations . . .8.2.1 One-sample Case . . . . . . . . . . .8.2.2 Two-sample Case . . . . . . . . . . .8.2.3 Problems . . . . . . . . . . . . . . .8.3 Test for Variances of Normal Populations . .8.3.1 One-sample Case . . . . . . . . . . .8.3.2 Two-sample Case . . . . . . . . . . .8.3.3 Problems . . . . . . . . . . . . . . .8.4 Tests for Binomial Populations . . . . . . .8.4.1 One-sample Case . . . . . . . . . . .8.4.2 Two-sample Case . . . . . . . . . . .8.4.3 Problems . . . . . . . . . . . . . . .8.5 Most Powerful and Uniformly Most PowerfulProblems . . . . . . . . . . . . . . . . . . . .8.6 Likelihood Ratio Tests . . . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . .8.7 Chi-Square Tests for Count Data . . . . . .8.7.1 Chi-Square Goodness-of-Fit Test . .8.7.2 Chi-Square Test of Independence . .8.7.3 Problems . . . . . . . . . . . . . . .8.8 Kolmogorov–Smirnov Test . . . . . . . . . .Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Nonparametric Estimation9.1 Kernel Density Estimation . . . . . . . . . . .9.1.1 Basic Ideas . . . . . . . . . . . . . . .9.1.2 Case Study 1: Rectangular Kernel . . .9.1.3 Case Study 2: Triangular and Gaussian9.1.4 Problems . . . . . . . . . . . . . . . .9.2 Order Statistics . . . . . . . . . . . . . . . . .9.2.1 Basics of Order Statistics . . . . . . . .9.2.2 Two Important Case Studies . . . . . .9.2.3 Joint Distribution of Order Statistics .Problems . . . . . . . . . . . . . . . . . . . . .Copyright 2018 ACTEX Learning. . . . . . . . . . . . .Kernels. . . . . . . . . . . . . . . . . . . . . . . . 7832841843ACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
xiiIIICONTENTSExtended Linear Models86710 Linear Regression Models10.1 Model Formulation and Parameter Estimation . . . . . . . . . . . .10.2 Model Construction . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2.1 Types of Explanatory Variables . . . . . . . . . . . . . . . .10.2.2 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 Statistical Inference . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3.1 Properties of Least Squares Estimators . . . . . . . . . . . .10.3.2 t-test for the Significance of a Single Regression Coefficient .10.3.3 F -test for the Significance of Several Regression Coefficients10.4 ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4.1 One-factor ANOVA . . . . . . . . . . . . . . . . . . . . . . .10.4.2 Two-factor ANOVA . . . . . . . . . . . . . . . . . . . . . . .10.5 Estimation and Prediction . . . . . . . . . . . . . . . . . . . . . . .10.5.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6 Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6.1 Influential Points . . . . . . . . . . . . . . . . . . . . . . . .10.6.2 Other Diagnostic Tools . . . . . . . . . . . . . . . . . . . . .10.7 End-of-chapter Problems . . . . . . . . . . . . . . . . . . . . . . . .11 Statistical Learning11.1 A Primer on Statistical Learning . . . . . . . .11.1.1 Fundamental Concepts . . . . . . . . . .11.1.2 Assessing the Accuracy of a Model . . .11.2 Resampling Methods . . . . . . . . . . . . . . .11.2.1 Cross Validation . . . . . . . . . . . . .11.2.2 Bootstrap . . . . . . . . . . . . . . . . .11.3 Variable Selection . . . . . . . . . . . . . . . . .11.3.1 Model Summary Statistics . . . . . . . .11.3.2 Best Subset Selection . . . . . . . . . . .11.3.3 Automatic Variable Selection Procedures11.4 Shrinkage Methods . . . . . . . . . . . . . . . .11.4.1 Ridge Regression . . . . . . . . . . . . .11.4.2 Lasso . . . . . . . . . . . . . . . . . . . .11.5 Dimension Reduction Techniques . . . . . . . .11.5.1 Principal Components Analysis . . . . .11.5.2 Partial Least Squares . . . . . . . . . . .11.6 Nonlinear Models . . . . . . . . . . . . . . . . .11.6.1 Spline Regression . . . . . . . . . . . . .11.6.2 Smoothing Splines . . . . . . . . . . . .11.6.3 Local Regression . . . . . . . . . . . . .11.6.4 Generalized Additive Models . . . . . . .Copyright 2018 ACTEX 10041004100710101010101110121012101510171018ACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
CONTENTSxiii11.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101912 Generalized Linear Models12.1 Exponential Family of Distributions . . . . .12.2 Estimation of Parameters . . . . . . . . . .12.3 Prediction . . . . . . . . . . . . . . . . . . .12.4 Assessing Model Fit by Deviance . . . . . .12.5 Hypothesis Tests on Regression Coefficients12.6 End-of-chapter Problems . . . . . . . . . . .13 Special GLMs for Insurance Data13.1 Logistic Regression . . . . . . . . . . . .13.2 Nominal and Ordinal Logistic Regression13.3 Poisson Regression . . . . . . . . . . . .13.4 End-of-chapter Problems . . . . . . . . .IV.1027102910371044104910551064.1081. 1083. 1102. 1108. 1118Time Series with Constant Variance113914 Introduction to Time Series Analysis114314.1 Trends and Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114314.2 Autocorrelation and Cross Correlation . . . . . . . . . . . . . . . . . . . . . 114914.3 End-of-chapter Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115715 ARIMA Models15.1 White Noise and Random Walk Processes15.2 Autoregressive Models . . . . . . . . . . .15.3 Moving Average Models . . . . . . . . . .15.4 ARMA Models . . . . . . . . . . . . . . .15.5 Non-stationary Models . . . . . . . . . . .15.6 End-of-chapter Problems . . . . . . . . . .V.Practice ExaminationsPractice Examination 1 . . . . . . .Solutions to Practice Examination 1Practice Examination 2 . . . . . . .Solutions to Practice Examination 2Practice Examination 3 . . . . . . .Solutions to Practice Examination 3Copyright 2018 ACTEX 1269129413081335ACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
xivCopyright 2018 ACTEX LearningCONTENTSACTEX Study Manual for CAS Exam MAS-ISpring 2018 EditionAmbrose Lo
Part IProbability Models(Stochastic Processes andSurvival Models)1
Chapter 1Poisson ProcessesLEARNING OBJECTIVES1. Understand and apply the properties of Poisson processes: For increments in the homogeneous caseFor interval times in the homogeneous caseFor increments in the non-homogeneous caseResulting from special types of events in the Poisson processResulting from sums of independent Poisson processesRange of weight: 0-5 percent2. For any Poisson process and the inter-arrival and waiting distributionsassociated with the Poisson process, calculate: Expected values Variances ProbabilitiesRange of weight: 0-5 percent3. For a compound Poisson process, calculate moments associated withthe value of the process at a given time.Range of weight: 0-5 percent4. Apply the Poisson Process concepts to calculate the hazard functionand related survival model concepts. Relationship between hazard rate, probability density function andcumulative distribution function Effect of memoryless nature of Poisson distribution on survivaltime estimationRange of weight: 2-8 percent3
4CHAPTER 1. POISSON PROCESSESChapter overview: As a prospective P&C actuary, you would be interested in monitoringthe number of insurance claims an insurance company receives as time goes by and howthese claims can be appropriately analyzed by means of sound statistical analysis. In ExamMAS-I, we shall learn one way of modeling the flow of insurance claims – the Poisson process.This part of the syllabus has two required readings:(1) A study note by J.W. DanielThe study note is precise and concise, introducing main results mostly without proofand supplementing its exposition with a few simple examples. It is suitable for afirst-time introduction to Poisson processes.(2) The book entitled Introduction to Probability Models by S.M. Ross.This is a textbook used by a number of college courses on elementary applied probability. It balances rigor and intuition, and presents the theory of Poisson processes at alevel that is much deeper than that in the study note by Daniel. In particular, it treatsthe conditional distribution of the arrival times as well as the interp
Here are the characteristics of a typical CAS multiple-choice exam: 1.The questions are almost always arranged in the same order as the topics in the exam syllabus, so Question 1 is very likely a Poisson process question and Question 40 is a time series question. This implicitly gives you a hint as to which topic an exam question is testing.
ACTEX P Study Manual, Spring 2018 Edition ACTEX is eager to provide you with helpful study material to assist you in gaining the necessary knowledge to become a successful actuary. In turn we would like your help in
ACTEX P Study Manual, Spring 2018 Edition, Second Printing ACTEX is eager to provide you with helpful study material to assist you in gaining the necessary knowledge to become a successful actuary. In turn we would like your help in
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5 CAS exam structure CAS 1 SOA P – Probability CAS 2 SOA FM – Financial Mathematics CAS 3L* SOA MLC – Life Contingencies CAS 3F SOA MFE – Financial Economics CAS 4 SOA C – Actuarial Modeling CAS 5 – Basic Ratemaking and Reserving CAS 6
Myers Park High School 2400 Colony Road Charlotte, NC 28209 www.myersparkib.org . Table of Contents: 1. Why do CAS . Service Strand 7. CAS Experiences 8. Planning CAS Experiences 9. CAS Project 10.CAS Learning Outcomes 11.CAS Portfolio 12.MPHS CAS Requirements and Guidelines 11th Grade (Class of 2022) Requirements 12th Grade (Class of 2021 .
10 tips och tricks för att lyckas med ert sap-projekt 20 SAPSANYTT 2/2015 De flesta projektledare känner säkert till Cobb’s paradox. Martin Cobb verkade som CIO för sekretariatet för Treasury Board of Canada 1995 då han ställde frågan
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