# Exam 3L

G. Stolyarov II
First Edition Published in August-October 2008
Second Edition Published in July 2014

 Section Page Section 1: Life-Table Probability Functions: Practice Problems and Solutions – Part 1 4 Section 2: Life-Table Probability Functions: Practice Problems and Solutions – Part 2 6 Section 3: Curate-Future-Lifetime 8 Section 4: Force of Mortality 10 Section 5: Identities Involving Life Table Functions and Force of Mortality 12 Section 6: Life-Table Probability Functions: Practice Problems and Solutions – Part 3 14 Section 7: Life-Table Probability Functions: Practice Problems and Solutions – Part 4 16 Section 8: Deterministic Survivorship Groups 19 Section 9: Computing Expected Values for Continuous Random Variables 21 Section 10: Complete-Expectation-of-Life 23 Section 11: Median Future Lifetime 25 Section 12: Computing Expected Values for Discrete Random Variables 28 Section 13: Life-Table Probability Functions: Practice Problems and Solutions – Part 5 31 Section 14: Life-Table Probability Functions: Practice Problems and Solutions – Part 6 33 Section 15: The Uniform-Distribution-of-Deaths Assumption for Fractional Ages 36 Section 16: The Uniform-Distribution-of-Deaths Assumption for Fractional Ages: Practice Problems and Solutions – Part 2 38 Section 17: Exam-Style Questions on Life-Table Functions – Part 1 40 Section 18: Exam-Style Questions on Life-Table Functions – Part 2 44 Section 19: Temporary Complete Life Expectancy 47 Section 20: Recursion Formulas 49 Section 21: Term Life Insurance 51 Section 22: Moments of the Present Value of a Term Life Insurance Policy 54 Section 23: Whole Life Insurance 57 Section 24: Makeham's Law of Mortality 60 Section 25: Weibull's Law of Mortality 63 Section 26: Probability Density Functions and Cumulative Distribution Functions of Present-Value Random Variables 65 Section 27: Pure Endowments 68 Section 28: Endowment Insurance 71 Section 29: Deferred Life Insurance 76 Section 30: Annually Increasing Whole Life Insurance 80 Section 31: Continuously Increasing Whole Life Insurance 84 Section 32: Annually Decreasing Term Life Insurance 88 Section 33: Insurances Payable at the End of the Year of Death 92 Section 34: Whole Life Insurance Payable at the End of the Year of Death 95 Section 35: Endowment Insurance Payable at the End of the Year of Death 98 Section 36: Exam-Style Questions on Life Insurance Policies 101 Section 37: Continuous Whole and Temporary Life Annuities 105 Section 38: Variances and Relationships Among Present Values of Continuous Whole and Temporary Life Annuities 108 Section 39: Continuous Deferred Life Annuities and Certain and Life Annuities 111 Section 40: Actuarial Accumulated Values and Derivatives of Actuarial Present Values of Continuous Life Annuities 115 Section 41: Discrete Whole Life Annuities 117 Section 42: Discrete Temporary Life Annuities 119 Section 43: Discrete Deferred Life Annuities and Certain and Life Annuities 121 Section 44: The Equivalence Principle and Fully Continuous Benefit Premiums for Whole, Term, and Endowment Life Insurance Policies 124 Section 45: Fully Continuous Benefit Premiums for Pure Endowments and Exam-Style Questions on Life Insurance and Life Annuities 128 Section 46: Exam Style Questions on Life Table Functions for Independent Lives and Multiple Causes of Decrement 133 Section 47: The Pareto and Weibull Probability Distributions 138 Section 48: The Negative Binomial Distribution and Some Associated Mixture Distributions 140 Section 49: The Lognormal Probability Distribution, Markov Chains, and Assorted Exam-Style Questions 142 Section 50: Probability Generating Functions, Poisson Processes, and Assorted Exam-Style Questions 145 Section 51: Binomial, Pareto, and Compound Probability Distributions 149 Section 52: Fully Discrete Benefit Premiums, Infinite Divisibility Property for Some Probability Distributions, and Assorted Exam-Style Questions 152 Section 53: Order Statistics, Likelihood Functions, and Maximum Likelihood Estimates 157 Section 54: Exam-Style Questions on the Method of Moments, Significance Levels, and the Neyman-Pearson Lemma 161 Section 55: Chi-Square Goodness-of-Fit Test 167 Section 56: Hypothesis Testing: Type I and Type II Errors, P-Values, and the Student-t Distribution 172 Section 57: Least-Squares Regression and Assorted Exam-Style Questions 176 Section 58: Fully Discrete Benefit Reserves 179 Section 59: The Central Limit Theorem and the Continuity Correction for Normal Approximations 183 Section 60: Characteristics of Estimators, Mean Square Error, and Assorted Exam Style Questions 186 Section 61: Residuals in Linear Regression and Assorted Exam-Style Questions 189 Section 62: Assorted Exam-Style Questions on Compound Poisson Processes, Normal Approximations, Force of Mortality, and Multiple Forces of Decrement 192 Section 63: Exam-Style Questions on Markov Chains and Percentile Calculations 196 Section 64: Poisson Processes and the Gamma Distribution 201 Section 65: The F Distribution and Hypothesis Testing of Variances 204 Section 66: Joint-Life Status and Last-Survivor Status 208 Section 67: Assorted Exam-Style Review Questions for Actuarial Exam 3L 211 Section 68: Challenging Exam-Style Review Questions for Actuarial Exam 3L 214 Section 69: Exam-Style Questions for Actuarial Exam 3L – Part 3 218 Section 70: Pooled Sample Variances and Covariances for Multiple-Life Statuses 222 Section 71: The Coefficient of Determination in Linear Regression, Fully Continuous Benefit Reserves Under DeMoivre's Law, and Assorted Exam-Style Questions 226 Section 72: Chi-Square Test for the Variance and Assorted Exam-Style Questions 230 Section 73: Exam-Style Questions for Actuarial Exam 3L – Part 4 234 Section 74: Exam-Style Questions for Actuarial Exam 3L – Part 5 238 Section 75: Exam-Style Questions for Actuarial Exam 3L – Part 6 240 Section 76: Exam-Style Questions for Actuarial Exam 3L – Part 7 243 Section 77: Exam-Style Questions for Actuarial Exam 3L – Part 8 246 About Mr. Stolyarov 248

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.

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