Exam P Flashcards exams. Key concepts. Important formulas. Efficient methods. Advice on exam technique

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1 Exam P Flashcards 01 exams Key concepts Important formulas Efficient methods Advice on exam technique

2 All study material produced by BPP Professional Education is copyright and is sold for the exclusive use of the purchaser. You may not hire out, lend, give out, store or transmit electronically or photocopy any part of the study material. You must take care of your study material to ensure that it is not used or copied by anybody else. Legal action will be taken if these terms are infringed. In addition, we may seek to take disciplinary action through the profession or through your employer.

3 CONTENTS Contents page How to use these flashcards Probability 3 Counting techniques 6 Conditional probability 8 Bayes Theorem 10 Discrete random variables 11 Common discrete distributions 14 Continuous random variables 18 Common continuous distributions 1 The normal distribution 4 Bivariate distributions 9 Conditional expectation and variance 35 The bivariate normal distribution 36 Transformations of random variables 37 Moment generating functions 43 Insurance concepts and terminology 46 BPP Professional Education: 01 exams Page 1

4 HOW TO USE THESE FLASHCARDS These flashcards are designed to help you to prepare efficiently in the run-up to the Course P exam of the Society of Actuaries. They include conceptual ideas, key formulas and techniques for efficient problem solving. Typical questions on a Course P examination require students to understand and apply several concepts in order to set up a solution, and then perform a series of computations to complete it. So don t look at the lists of formulas as simply being memorization work. There are often simple ideas that underlie the formulas as well as basic mathematical reasons why they are correct. Strive to understand and learn the key relations from this point of view and your knowledge will not be the superficial type that may collapse under the stress of taking the examination. The more that you understand conceptually, the easier it becomes to retain the key ideas and write them down quickly and accurately. Your understanding of probability concepts plays a huge role in Courses M and C that lie ahead where you will encounter more advanced and somewhat abstract probability concepts. You will need a solid foundation to be successful there. We have designed the flashcards so that they can be carried conveniently and read frequently in the final run-up to the exam, eg when sitting on a plane. We hope that you will personalize them by adding your own comments and notes, and checking each section when you feel confident with the material covered. You will probably also find these summaries useful when you are at the stage of working through the past exams. The BPP Exam P Question and Answer Bank contains a mixture of past exam questions and brand new exam-style questions, along with detailed solutions. By the time you have worked through these questions you will have a clear picture of what the exam is like and what you need to work on to get ready for it. As a final tuneup, try one of the BPP Course P practice exams, containing all new exam-style questions. Good luck with your studying. Page BPP Professional Education: 01 exams

5 COUNTING TECHNIQUES Another counting rule 4. The number of distinct arrangements of n objects of which n 1 are identical, n are identical,, and n r are n! identical is n! n! n!. 1 Example of probability with equally likely outcomes r A lottery randomly picks a winning combination of 6 numbers from the whole numbers 1 through 48. Each possible outcome is equally likely. The number of outcomes is: ,71, The probability that a ticket (ie a combination of 6 particular numbers) is a winning combination is 1/1,71,51. The probability that the winning combination contains exactly 3 single-digit numbers, event E, is computed as follows: The number of ways in which exactly 3 of the 9 single-digit numbers can be selected is: The number of ways in which exactly 3 of the 39 double-digit numbers can be selected is: , So the probability of event E is: 84 9,139 PrE ,71,51 BPP Professional Education: 01 exams Page 7

6 CONDITIONAL PROBABILITY Conditional probability The conditional probability of A given B is denoted PrA B. The conditional probability is defined to be: Pr AB PrA B where PrB Pr B 0 If we represent events A and B using a Venn diagram: A B w x y z then the main relationships are as follows: Pr A w x PrB x y Pr A B x Pr A B wxy 1 z Pr Pr Pr A B x AB Pr B x y Pr A B x BA Pr A w x Page 8 BPP Professional Education: 01 exams

THE NORMAL DISTRIBUTION The standard normal distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

7 THE NORMAL DISTRIBUTION The standard normal distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It has a probability density function, ( x), equal to: x 1 x exp for x The distribution function of the standard normal distribution is usually denoted by x : s x 1 x Pr( X x) e ds (shaded area below) This probability cannot be computed using the fundamental theorem of calculus since there is no closed form expression for the anti-derivative. Numerically approximated values of x for x 0, 0.1, 0.,, 3.0 can be found in tables that are distributed with the Course P examination. To compute values such as 0.3 you will need to use the symmetry of the standard normal pdf: x PrX x PrX x 1 x BPP Professional Education: 01 exams Page 5

8 THE NORMAL DISTRIBUTION Properties of the normal distribution Property 1, X 0,1 X N N This property allows us to use the standard normal distribution tables to calculate probabilities for any normal distribution. For example, suppose that X N,, then: Pr Pr a a X b X b b a Property X N,, Y axb Y N ab, a Property 3 X1,, Xn independent and Xi N i, i n n ax 1 1 ax n n N ai i, ai i i1 i1 Property 4 X1,, Xn independent and Xi N, X1 X n X X N, and N0,1 n n / n The average of a set of observations from a normal distribution is itself distributed normally. The variance of this average reduces as the sample size (n) increases, ie the greater the sample size, the more confident we can be that the mean of the observations is close to the mean of the underlying normal distribution. Page 6 BPP Professional Education: 01 exams

9 THE MOMENT GENERATING FUNCTION Generating functions of common distributions 1. Uniform M t bt at e e t b a t. Normal M t exp t 3. Gamma M t 4. Exponential M t 1 t 1 1 t 1 t 1 1 t t 5. Poisson M t exp e 1 6. Binomial M t pe t q n 1 qe 7. Negative binomial M t p t r 1 qe 8. Geometric M t p t 1 BPP Professional Education: 01 exams Page 45

10 INSURANCE CONCEPTS AND TERMINOLOGY How insurance works Insurance involves two parties: the insurance company (the insurer) the policyholder. The policyholder purchases an insurance policy by paying a premium to the insurance company. The premium is calculated by the insurance company and reflects the level of risk. In return for the premium, the insurer promises to pay the policyholder an amount of money (called a benefit or claim payment), if the policyholder were to suffer a loss due to a specified event within a specified time period. The insurance company can afford to carry the risk because it sells a large number of similar policies. (The standard deviation of a sum of n independent variables distributed like X is whereas the mean is nex.) n X, Each policyholder pays a premium greater than the expected benefit amount to be received. The actuary needs to consider two aspects of the claims process: claim frequency (how many claims will occur ) claim severity (how large is a claim likely to be) Terminology If there is no deductible or policy limit, then the claim payment will be equal to the loss amount suffered by the policyholder. If an insurance policy has a deductible d and X is a policyholder s loss, the claim payment will be X d if X d. Nothing is paid by the insurer if the loss satisfies X d. If an insurance policy has a limit L and X is a policyholder s loss, the claim payment will be equal to X if the loss is less than the limit L. However, if the loss exceeds the limit, then the claim payment will be equal to the limit. The net premium or pure premium is the expected claim payment. Page 46 BPP Professional Education: 01 exams

PROBABILITY. Wiley. With Applications and R ROBERT P. DOBROW. Department of Mathematics. Carleton College Northfield, MN

PROBABILITY. Wiley. With Applications and R ROBERT P. DOBROW. Department of Mathematics. Carleton College Northfield, MN PROBABILITY With Applications and R ROBERT P. DOBROW Department of Mathematics Carleton College Northfield, MN Wiley CONTENTS Preface Acknowledgments Introduction xi xiv xv 1 First Principles 1 1.1 Random