The
Rational Argumentator
Free Tools for Rational Education
The Actuary's Free Study Guide for
Exam 4 / Exam C
First Edition Published in July-October 2009
Second Edition Published in July 2014
Recommend this page.
Click here to download a free PDF copy of the Second Edition of this study guide.
Table of Contents
|
Section |
Page |
|
Section 1: Raw Moments, Central
Moments, Excess Loss Variables, and Left-Censored and Shifted Variables |
5 |
|
Section 2: The
Empirical Model and Limited Loss Variables |
9 |
|
Section 3: Coefficient
of Variation, Skewness, Kurtosis, and a Shortcut Formula for Mean Excess Loss
Functions |
12 |
|
Section 4: Percentiles,
Value-at-Risk, and Tail-Value-at-Risk |
15 |
|
Section 5: The
Central Limit Theorem and Applications to Special Distributions |
18 |
|
Section 6: Moment
Generating Functions and Probability Generating Functions |
22 |
|
Section 7: Hazard
Rates and Light and Heavy Tails of Distributions |
24 |
|
Section 8: The
Equilibrium Distribution, Mean Excess Loss Functions, and Tails of
Distributions |
27 |
|
Section 9: Risk
Measures and Coherence |
29 |
|
Section 10: Parametric
Distributions and Scale Parameters |
32 |
|
Section 11: K-Point
Mixtures, Variable-Component Mixture Distributions, and Data-Dependent
Distributions |
35 |
|
Section 12: Exam-Style
Questions on the Central Limit Theorem, Skewness, Percentile Matching, and
Ogives |
38 |
|
Section 13: Distributions
of Multiples and Powers of Random Variables |
42 |
|
Section 14: The
Gamma Function, Incomplete Gamma Function, and Exponentiation of Random
Variables |
44 |
|
Section 15: Mixture
Distributions and Some Review Questions |
46 |
|
Section 16: Mixture
Distributions Involving Discrete Random Variables |
49 |
|
Section 17: Frailty
Models |
52 |
|
Section 18: Spliced
Distributions |
54 |
|
Section 19: The
Linear Exponential Family of Distributions |
58 |
|
Section 20: Properties
of Poisson, Negative Binomial, and Geometric Distributions |
60 |
|
Section 21: Properties
of the Binomial Distribution and the (a, B, 0) Class of Distributions |
63 |
|
Section 22: The
(a, B, 1) Class of Distributions |
65 |
|
Section 23: The
Logarithmic Distribution |
67 |
|
Section 24: Assorted
Exam-Style Questions for Exam 4 / Exam C – Part 1 |
69 |
|
Section 25: Applications
of Ordinary Deductibles Per Payment and Per Loss |
74 |
|
Section 26: Applications
of Franchise Deductibles Per Payment and Per Loss |
78 |
|
Section 27: Loss
Elimination Ratios and Inflation Effects for Ordinary Deductibles |
81 |
|
Section 28: Policy
Limits and Associated Effects of Inflation |
83 |
|
Section 29: Coinsurance
and Treatment Thereof in Combination with Ordinary Deductibles, Policy
Limits, and Inflation |
85 |
|
Section 30: Impact
of a Deductible on the Number of Payments |
90 |
|
Section 31: Assorted
Exam-Style Questions for Exam 4 / Exam C – Part 2 |
93 |
|
Section 32: The
Collective Risk Model and the Individual Risk Model |
97 |
|
Section 33: Mean,
Variance, Third Central Moment, and Probability Calculations Pertaining to
Aggregate Loss Random Variables |
100 |
|
Section 34: Stop-Loss
Insurance and the Net Stop-Loss Premium |
103 |
|
Section 35: Moment
Generating Functions of Aggregate Random Variables and Probability
Calculations for Aggregate Random Variables with Exponential Severity |
106 |
|
Section 36: Using
Convolutions to Determine the Probability Distribution of Aggregate Random
Variables |
110 |
|
Section 37: The
Method of Rounding or Mass Dispersal |
113 |
|
Section 38: The
Method of Local Moment Matching |
116 |
|
Section 39: Effects
of Individual Policy Modifications on the Aggregate Payment Random Variable |
120 |
|
Section 40: Properties
of Aggregate Losses in the Individual Risk Model |
123 |
|
Section 41: Parametric
Approximations of Probability Distributions of Aggregate Losses in the
Individual Risk Model |
127 |
|
Section 42: The
Empirical Distribution Function, the Cumulative Hazard Rate Function, and the
Nelson-Åalen Estimate |
131 |
|
Section 43: Kernel
Smoothed Distributions, Ogives, and Histograms |
134 |
|
Section 44: Exam-Style
Questions on the Smoothed Empirical Estimate, the Nelson-Åalen Estimate, and
Aggregate Random Variables |
139 |
|
Section 45: Properties
of Estimators |
143 |
|
Section 46: Interval
Estimators and Hypothesis Testing |
146 |
|
Section 47: Modifications
of Observations and Calculations Pertaining to the Risk Set for Modified
Observations |
150 |
|
Section 48: The
Kaplan-Meier Product-Limit Estimator |
155 |
|
Section 49: Full
Credibility of Data in Limited Fluctuation Credibility Theory |
160 |
|
Section 50: Estimating
the Variances of Empirical Survival Functions and Empirical Probability
Estimates |
163 |
|
Section 51: Greenwood's
Approximation and Variances of Products of Random Variables |
166 |
|
Section 52: Log-Transformed
Confidence Intervals for Survival Functions and Cumulative Hazard Rate
Functions |
170 |
|
Section 53: Kernel
Density Estimators and Uniform, Triangular, and Gamma Kernels |
175 |
|
Section 54: Partial
Credibility in Limited Fluctuation Credibility Theory |
178 |
|
Section 55: The
Kaplan-Meier Approximation for Large Data Sets and for Multiple Decrements |
183 |
|
Section 56: The
Method of Moments and Maximum Likelihood Estimation |
188 |
|
Section 57: Exam-Style
Questions on Maximum Likelihood Estimation |
193 |
|
Section 58: Fisher
Information for One Parameter |
197 |
|
Section 59: Fisher
Information for Multiple Parameters |
200 |
|
Section 60: The
Delta Method for Estimating Functions of Parameters |
204 |
|
Section 61: Non-Normal
Confidence Intervals for Parameters |
208 |
|
Section 62: Prior,
Model, Marginal, Posterior, and Predictive Distributions in Bayesian
Estimation |
211 |
|
Section 63: Loss
Functions and the Expected Value of the Predictive Distribution in Bayesian
Estimation |
214 |
|
Section 64: Credibility
Intervals and the Bayesian Central Limit Theorem in Bayesian Estimation |
218 |
|
Section 65: Conjugate
Prior Distributions in Bayesian Estimation |
222 |
|
Section 66: Exam-Style
Questions on Bayesian Estimation |
225 |
|
Section 67: Exam-Style
Questions on Bayesian Estimation – Part 2 |
228 |
|
Section 68: Exam-Style
Questions on Bühlmann Credibility |
231 |
|
Section 69: Exam-Style
Questions on Bühlmann Credibility – Part 2 |
236 |
|
Section 70: Exam-Style
Questions on Bühlmann-Straub Credibility |
240 |
|
Section 71: Distribution
and Probability Density Functions for Modified Data, Difference Functions,
and the Kolmogorov-Smirnov Test |
245 |
|
Section 72: Exam-Style
Questions on the Kolmogorov-Smirnov Test and Bühlmann Credibility |
250 |
|
Section 73: Nonparametric
and Semiparametric Empirical Bayes Estimation of Bühlmann And Bühlmann-Straub
Credibility Factors |
254 |
|
Section 74: Exam-Style
Questions on Nonparametric Empirical Bayes Estimation of Bühlmann and
Bühlmann-Straub Credibility Factors |
259 |
|
Section 75: Exam-Style
Questions on Semiparametric Empirical Bayes Estimation of Bühlmann and
Bühlmann-Straub Credibility Factors and the Chi-Square Goodness of Fit Test |
264 |
|
Section 76: The
Anderson-Darling Test |
270 |
|
Section 77: The
Likelihood Ratio Test |
274 |
|
Section
78: The Schwarz Bayesian Criterion and Exam-Style Questions on Hypothesis
Testing of Models |
279 |
|
Section 79: Basics
of Simulation and the Inversion Method |
283 |
|
Section 80: Exam-Style
Questions on the Inversion Method |
288 |
|
Section 81: Exam-Style
Questions on Simulation |
292 |
|
Section 82: Estimates
for Value-at-Risk and Tail-Value-at-Risk for Simulated Samples |
296 |
|
Section 83: The
Bootstrap Method for Estimating Mean Square Error |
299 |
|
Section 84: Simulations
of Cumulative Distribution Functions and Exam-Style Questions on Simulation |
303 |
|
Section 85: Assorted
Exam-Style Questions for Exam 4/C – Part 3 |
307 |
|
Section 86: Assorted
Exam-Style Questions for Exam 4/C – Part 4 |
311 |
|
Section 87: The
Life Table Methodology for Estimating Risk Sets and Assorted Exam-Style
Questions for Exam 4/C |
316 |
|
Section 88: Assorted
Exam-Style Questions for Exam 4/C – Part 5 |
320 |
|
Section 89: Assorted
Exam-Style Questions for Exam 4/C – Part 6 |
323 |
|
Section 90: Assorted
Exam-Style Questions for Exam 4/C – Part 7 |
327 |
|
Section 91: Assorted
Exam-Style Questions for Exam 4/C – Part 8 |
330 |
|
Section 92: Assorted
Exam-Style Questions for Exam 4/C – Part 9 |
333 |
|
Section 93: The
Continuity Correction for Normal Approximations and Exam-Style Questions on
Mixed and Aggregate Distributions |
336 |
|
Section 94: Assorted
Exam-Style Questions for Exam 4/C – Part 10 |
339 |
|
Section 95: Assorted
Exam-Style Questions for Exam 4/C – Part 11 |
342 |
|
Section 96: Assorted
Exam-Style Questions for Exam 4/C – Part 12 |
346 |
|
Section 97: Assorted
Exam-Style Questions for Exam 4/C – Part 13 |
349 |
|
Section 98: Assorted
Exam-Style Questions for Exam 4/C – Part 14 |
352 |
|
Section 99: Assorted
Exam-Style Questions for Exam 4/C – Part 15 |
356 |
|
Section 100: Assorted
Exam-Style Questions for Exam 4/C – Part 16 |
359 |
|
Section 101: Assorted
Exam-Style Questions for Exam 4/C – Part 17 |
362 |
|
Section 102: Assorted
Exam-Style Questions for Exam 4/C – Part 18 |
365 |
|
Section 103: Assorted
Exam-Style Questions for Exam 4/C – Part 19 |
370 |
|
Section 104: Assorted
Exam-Style Questions for Exam 4/C – Part 20 |
374 |
|
Section 105: Assorted
Exam-Style Questions for Exam 4/C – Part 21 |
378 |
|
Section 106: Assorted
Exam-Style Questions for Exam 4/C – Part 22 |
382 |
|
Section 107: Assorted
Exam-Style Questions for Exam 4/C – Part 23 |
387 |
|
Section 108: Assorted
Exam-Style Questions for Exam 4/C – Part 24 |
393 |
|
Section 109: Assorted
Exam-Style Questions for Exam 4/C – Part 25 |
398 |
|
Section 110: Assorted
Exam-Style Questions for Exam 4/C – Part 26 |
402 |
|
Section 111: Assorted
Exam-Style Questions for Exam 4/C – Part 27 |
407 |
|
Section 112: Assorted
Exam-Style Questions for Exam 4/C – Part 28 |
411 |
|
Section 113: Assorted
Exam-Style Questions for Exam 4/C – Part 29 |
415 |
|
Section 114: Assorted
Exam-Style Questions for Exam 4/C – Part 30 |
418 |
|
Section 115: Assorted
Exam-Style Questions for Exam 4/C – Part 31 |
422 |
|
Section 116: Assorted
Exam-Style Questions for Exam 4/C – Part 32 |
426 |
|
About Mr.
Stolyarov |
430 |
Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress.
In December 2013, Mr. Stolyarov published Death is Wrong, an ambitious children’s book on life extension illustrated by his wife Wendy. Death is Wrong can be found on Amazon in paperback and Kindle formats.
Mr. Stolyarov has contributed articles to the Institute for Ethics and Emerging Technologies (IEET), The Wave Chronicle, Le Quebecois Libre, Brighter Brains Institute, Immortal Life, Enter Stage Right, Rebirth of Reason, The Liberal Institute, and the Ludwig von Mises Institute. Mr. Stolyarov also published his articles on Associated Content (subsequently the Yahoo! Contributor Network) from 2007 until its closure in 2014, in an effort to assist the spread of rational ideas. He held the highest Clout Level (10) possible on the Yahoo! Contributor Network and was one of its Page View Millionaires, with over 3.1 million views.
Mr. Stolyarov holds the professional insurance designations of Associate of the Society of Actuaries (ASA), Associate of the Casualty Actuarial Society (ACAS), Member of the American Academy of Actuaries (MAAA), Chartered Property Casualty Underwriter (CPCU), Associate in Reinsurance (ARe), Associate in Regulation and Compliance (ARC), Associate in Personal Insurance (API), Associate in Insurance Services (AIS), Accredited Insurance Examiner (AIE), and Associate in Insurance Accounting and Finance (AIAF).
Mr. Stolyarov has written a science fiction novel, Eden against the Colossus, a philosophical treatise, A Rational Cosmology, a play, Implied Consent, and a free self-help treatise, The Best Self-Help is Free. You can watch his YouTube Videos. Mr. Stolyarov can be contacted at gennadystolyarovii@gmail.com.
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Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.Read Mr. Stolyarov's comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, here.