# Exam 4 / Exam C

G. Stolyarov II
First Edition Published in July-October 2009
Second Edition Published in July 2014

 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 RightRebirth 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.

This TRA feature has been edited in accordance with TRA’s Statement of Policy.