Mahlerʼs Guide to Frequency Distributions Exam C prepared by Howard C. Mahler,

Mahlerʼs Guide to Frequency Distributions Exam C prepared by Howard C. Mahler, FCAS Copyright 2016 by Howard C. Mahler. Study Aid 2016-C-1 Howard Mahler hmahler@mac.com www.howardmahler.com/Teaching Mahlerʼs Guide to Frequency Distributions Copyright 2016 by Howard C. Mahler. Information in bold or sections whose title is in bold are more important for passing the exam. Larger bold type indicates it is extremely important. Information presented in italics (or sections whose title is in italics) should not be needed to directly answer exam questions and should be skipped on first reading. It is provided to aid the readerʼs overall understanding of the subject, and to be useful in practical applications. Highly Recommended problems are double underlined. Recommended problems are underlined. 1 Solutions to the problems in each section are at the end of that section. Section # Pages Section Name A 1 4 Introduction 2 5-15 Basic Concepts 3 16-41 Binomial Distribution 4 42-74 Poisson Distribution B 5 75-96 Geometric Distribution 6 97-122 Negative Binomial Distribution 7 123-150 Normal Approximation 151-163 C 8 151-163 Skewness 9 164-179 Probability Generating Functions 10 180-192 Factorial Moments 11 193-214 (a, b, 0) Class of Distributions 12 215-228 Accident Profiles D 13 229-252 Zero-Truncated Distributions 14 253-274 Zero-Modified Distributions 15 275-289 Compound Frequency Distributions 16 290-310 Moments of Compound Distributions 17 311-356 Mixed Frequency Distributions E 18 357-368 Gamma Function 19 369-411 Gamma-Poisson Frequency Process 20 412-422 Tails of Frequency Distributions F 21 423-430 Important Formulas and Ideas 1 Note that problems include both some written by me and some from past exams. The latter are copyright by the CAS and SOA, and are reproduced here solely to aid students in studying for exams. The solutions and comments are solely the responsibility of the author; the CAS and SOA bear no responsibility for their accuracy. While some of the comments may seem critical of certain questions, this is intended solely to aid you in studying and in no way is intended as a criticism of the many volunteers who work extremely long and hard to produce quality exams. In some cases Iʼve rewritten these questions in order to match the notation in the current Syllabus. 2016-C-1, Frequency Distributions HCM 10/21/15, Page 1 Past Exam Questions by Section of this Study Aid2 Course 3 Course 3 Course 3 Course 3 Course 3 Course 3 CAS 3 SOA 3 CAS 3 Section Sample 5/00 11/00 5/01 11/01 11/02 11/03 11/03 5/04 1 2 3 14 4 16 5 6 18 7 8 28 9 10 11 25 28 32 12 13 14 37 15 16 2 16 36 30 27 26 17 13 18 19 12 4 3 15 27 5 15 20 The CAS/SOA did not release the 5/02 and 5/03 exams. From 5/00 to 5/03, the Course 3 Exam was jointly administered by the CAS and SOA. Starting in 11/03, the CAS and SOA gave separate exams. (See the next page.) 2 Excluding any questions that are no longer on the syllabus. 2016-C-1, Frequency Distributions HCM 10/21/15, Page 2 CAS 3 SOA 3 CAS 3 SOA M CAS 3 SOA M CAS 3 CAS 3 SOA M Section 11/04 11/04 5/05 5/05 11/05 11/05 5/06 11/06 11/06 1 2 3 22 24 8 15 4 23 39 24 32 5 6 21 28 32 23 24 31 22 7 8 9 10 25 11 16 19 31 12 13 14 15 27 16 18 35 30 17 32 19 39 18 19 10 20 4/C Section 5/07 The SOA did not release its 5/04 and 5/06 exams. This material was moved to Exam 4/C in 2007. The CAS/SOA did not release the 11/07 and subsequent exams. 2016-C-1, Frequency Distributions HCM 10/21/15, Page 3 Section 1, Introduction This Study Aid will review what a student needs to know about the frequency distributions in Loss Models. Much of the first seven sections you should have learned on Exam P. In actuarial work, frequency distributions are applied to the number of losses, the number of claims, the number of accidents, the number of persons injured per accident, etc. Frequency Distributions are discrete functions on the nonnegative integers: 0, 1, 2, 3, ... There are three named frequency distributions you should know: Binomial, with special case Bernoulli Poisson Negative Binomial, with special case Geometric. Most of the information you need to know about each of these distributions is shown in Appendix B, attached to the exam. Nevertheless, since they appear often in exam questions, it is desirable to know these frequency distributions well, particularly the Poisson Distribution. In addition, one can make up a frequency distribution. How to work with such unnamed frequency distributions is discussed in the next section. In later sections, the important concepts of Compound Distributions and Mixed Distributions will be discussed. 3 The most important case of a mixed frequency distribution is the Gamma-Poisson frequency process. 3 Compound Distributions are mathematically equivalent to Aggregate Distributions, which are discussed in “Mahlerʼs Guide to Aggregate Distributions.” 2016-C-1, Frequency Distributions, §1 Introduction HCM 10/21/15, Page 4 Section 2, Basic Concepts The probability density function 4 f(i) can be non-zero at either a finite or infinite number of points. In the former case, the probability density function is determined by a table of its values at these finite number of points. The f(i) can take on any values provided they satisfy 0 ≤ f(i) ≤1 and f(i) i=0 ∞ ∑ = 1. For example: Number Probability Cumulative of Claims Density Function Distribution Function 0 0.1 0.1 1 0.2 0.3 2 0 0.3 3 0.1 0.4 4 0 0.4 5 0 0.4 6 0.1 0.5 7 0 0.5 8 0 0.5 9 0.1 0.6 10 0.3 0.9 11 0.1 1 Sum 1 The Distribution Function 5 is the cumulative sum of the probability density function: F(j) = f(i) i=0 j ∑ . In the above example, F(3) = f(0) + f(1) + f(2) + f(3) = 0.1 + 0.2 + 0 + 0.1 = 0.4. 4 Loss Models calls the probability density function of frequency the “probability function” or p.f. and uses the notation pk for f(k), the density at k. 5 Also called the cumulative distribution function. 2016-C-1, Frequency Distributions, §2 Basic Concepts HCM 10/21/15, Page 5 Moments: One can calculate the moments of such a distribution. For example, the first moment or mean is: (0)(0.1) + (1)(0.2) + (2)(0) + (3)(0.1) + (4)(0) + (5)(0) + (6)(0.1) + (7)(0) + (8)(0) + (9)(0.1) + (10)(0.3) + (11)(0.1) = 6.1. Probability x Number Probability Probability x Square of of Claims Density Function # of Claims # of Claims 0 0.1 0 0 1 0.2 0.2 0.2 2 0 0 0 3 0.1 0.3 0.9 4 0 0 0 5 0 0 0 6 0.1 0.6 3.6 7 0 0 0 8 0 0 0 9 0.1 0.9 8.1 10 0.3 3 30 11 0.1 1.1 12.1 Sum 1 6.1 54.9 E[X] = Σ i f(i) = Average of X = 1st moment about the origin = 6.1. E[X2] = Σ i2 f(i) = Average of X2 = 2nd moment about the origin = 54.9. The second moment is: (02)(0.1) + (12)(0.2) + (22)(0) + (32)(0.1) + (42)(0) + (52)(0) + (62)(0.1) + (72)(0) + (82)(0) + (92)(0.1) + (102)(0.3) + (112)(0.1) = 54.9. Mean = E[X] = 6.1. Variance = second central moment = E[(X - E[X])2] = E[X2] - E[X]2 = 17.69. Standard Deviation = Square Root of Variance = 4.206. The mean is the average or expected value of the random variable. For the above example, the mean is 6.1 claims. 2016-C-1, Frequency Distributions, §2 Basic Concepts HCM 10/21/15, Page 6 In general means add; E[X+Y] = E[X] + E[Y]. Also multiplying a variable by a constant multiplies the mean by the same constant; E[kX] = kE[X]. The mean is a linear operator: E[aX + bY] = aE[X] + bE[Y]. The mean of a frequency distribution can also be computed as a sum of its survival functions: 6 E[X] = Prob[X >i] i=0 ∞ ∑ = {1 - F(i)} i=0 ∞ ∑ . Mode and Median: The mean differs from the mode which represents the value most likely to occur. The mode is the point at which the density function reaches its maximum. The mode for the above example is 10 claims. For a discrete distribution, take the 100pth percentile as the first value at which F(x) ≥ p. 7 The 80th percentile for the above example is 10; F(9) = 0.6, F(10) = 0.9. The median is the 50th percentile. For frequency distributions, and other discrete distributions, the median is the first value at which the distribution function is greater than or equal to 0.5. The median for the above example is 6 claims; F(6) = 0.5. Definitions: Exposure Base: The basic unit of measurement upon which premium is determined. For example, the exposure base could be car-years, $100 of payrolls, number of insured lives, etc. The rate for Workersʼ Compensation Insurance might uploads/Philosophie/ guide-c.pdf

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