Study Manual for Exam P Exam 1 KSU Faculty

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TO OUR READERS, Please check A S M s web site at www studymanuals com for errata and. updates If you have any comments or reports of errata please. e mail us at mail studymanuals com or e mail Dr Krzysztof Ostaszewski. at krzysio krzysio net His web site is www krzysio net. The author would like to thank the Society of Actuaries and the Casualty Actuarial Society for. their kind permission to publish questions from past actuarial examinations. Copyright 2004 2012 by Krzysztof Ostaszewski All rights reserved. Reproduction in whole or in part without express written permission from the. author is strictly prohibited,INTRODUCTION, Before you start studying for actuarial examinations you need to familiarize yourself with the. Fundamental Rule for Passing Actuarial Examinations. You should greet every problem you see when you are taking the exam with these words Been. there done that, I will refer to this Fundamental Rule as the BTDT Rule If you do not follow the BTDT Rule. neither this manual nor any book nor any tutorial will be of much use to you And I do want to. help you so I must beg you to follow the BTDT Rule Allow me now to explain its meaning. If you are surprised by any problem on the exam you are likely to miss that problem Yet the. difference between a 5 and a 6 is one problem This surprise problem has great marginal value. There is simply not enough time to think on the exam Thinking is always the last resort on an. actuarial exam You may not have seen this very problem before but you must have seen a. problem like it before If you have not you are not prepared. If you have not thoroughly studied all topics covered on the exam you are taking you must have. subconsciously wished to spend more time studying a half a year s or even a year s worth. more But clearly the biggest reward for passing an actuarial examination is not having to take. it again By spending extra hours days or even weeks studying and memorizing all topics. covered on the exam you are saving yourself possibly as much as a year s worth of your life. Getting a return of one year on an investment of one day is better than anything you will ever. make on Wall Street or even lotteries unearned wealth is destructive thus objects called. lottery winnings are smaller than they appear Please study thoroughly without skipping any. topic or any kind of a problem To paraphrase my favorite quote from Ayn Rand for zat you. will be very grateful to yourself, I had once seen Harrison Ford being asked what he answers to people who tell him May the. Force be with you He said Force Yourself That s what you need to do You have much. more will than you assume so go make yourself study. Krzys Ostaszewski,Bloomington Illinois September 2004.
P S I want to thank my wife Patricia for her help and encouragement in writing of this manual. I also want to thank Hal Cherry for his help and encouragement Any errors in this work are. mine and mine only If you find any please be kind to let me know about them because I. definitely want to correct them, ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2011 by Krzysztof Ostaszewski 1. TABLE OF CONTENTS,INTRODUCTION 1,SECTION 1 GENERAL PROBABILITY 4. SECTION 2 RANDOM VARIABLES AND,PROBABILITY DISTRIBUTIONS 22. SECTION 3 MULTIVARIATE DISTRIBUTIONS 57,SECTION 4 RISK MANAGEMENT AND INSURANCE 92. PRACTICE EXAMINATIONS AN INTRODUCTION 109,PRACTICE EXAMINATION NUMBER 1 110.
PRACTICE EXAMINATION NUMBER 1 SOLUTIONS 118,PRACTICE EXAMINATION NUMBER 2 141. PRACTICE EXAMINATION NUMBER 2 SOLUTIONS 149,PRACTICE EXAMINATION NUMBER 3 172. PRACTICE EXAMINATION NUMBER 3 SOLUTIONS 179,PRACTICE EXAMINATION NUMBER 4 199. PRACTICE EXAMINATION NUMBER 4 SOLUTIONS 207,PRACTICE EXAMINATION NUMBER 5 228. PRACTICE EXAMINATION NUMBER 5 SOLUTIONS 236,PRACTICE EXAMINATION NUMBER 6 256.
PRACTICE EXAMINATION NUMBER 6 SOLUTIONS 263,PRACTICE EXAMINATION NUMBER 7 289. PRACTICE EXAMINATION NUMBER 7 SOLUTIONS 296,PRACTICE EXAMINATION NUMBER 8 315. PRACTICE EXAMINATION NUMBER 8 SOLUTIONS 322,PRACTICE EXAMINATION NUMBER 9 343. PRACTICE EXAMINATION NUMBER 9 SOLUTIONS 351, ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2012 by Krzysztof Ostaszewski 2. PRACTICE EXAMINATION NUMBER 10 374,PRACTICE EXAMINATION NUMBER 10 SOLUTIONS 381.
PRACTICE EXAMINATION NUMBER 11 400,PRACTICE EXAMINATION NUMBER 11 SOLUTIONS 408. PRACTICE EXAMINATION NUMBER 12 433,PRACTICE EXAMINATION NUMBER 12 SOLUTIONS 440. PRACTICE EXAMINATION NUMBER 13 463,PRACTICE EXAMINATION NUMBER 13 SOLUTIONS 470. PRACTICE EXAMINATION NUMBER 14 493,PRACTICE EXAMINATION NUMBER 14 SOLUTIONS 500. PRACTICE EXAMINATION NUMBER 15 524,PRACTICE EXAMINATION NUMBER 15 SOLUTIONS 531.
PRACTICE EXAMINATION NUMBER 16 551,PRACTICE EXAMINATION NUMBER 16 SOLUTIONS 557. PRACTICE EXAMINATION NUMBER 17 578,PRACTICE EXAMINATION NUMBER 17 SOLUTIONS 585. PRACTICE EXAMINATION NUMBER 18 606,PRACTICE EXAMINATION NUMBER 18 SOLUTIONS 614. PRACTICE EXAMINATION NUMBER 19 635,PRACTICE EXAMINATION NUMBER 19 SOLUTIONS 643. PRACTICE EXAMINATION NUMBER 20 666,PRACTICE EXAMINATION NUMBER 20 SOLUTIONS 673.
ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2012 by Krzysztof Ostaszewski 3. SECTION 1 GENERAL PROBABILITY,Basic probability concepts. Probability concepts are defined for elements and subsets of a certain set a universe under. consideration That universe is called the probability space or sample space It can be a finite. set a countable set a set whose elements can be put in a sequence in fact a finite set is also. countable or an infinite uncountable set e g the set of all real numbers or any interval on the. real line Subsets of a given probability space for which probability can be calculated are called. events An event represents something that can possibly happen In the most general probability. theory not every set can be an event But this technical issue does not come up on any lower. level actuarial examinations It should be noted that while not all subsets of a probability space. must be events it is always the case that the empty set is an event the entire space S is an event. a complement of an event is an event and a set theoretic union of any sequence of events is an. event The entire probability space will be usually denoted by S always in this text or in. more theoretical probability books It encompasses everything that can possibly happen. The simplest possible event is a one element set If we perform an experiment and observe what. happens as its outcome such a single observation is called an elementary event Another name. commonly used for such an event is a sample point, A union of two events is an event which combines all of their elements regardless of whether. they are common to those events or not For two events A and B their union is denoted by. A B For a more general finite collection of events A1 A2 An their union consists of all. elementary events that belong to any one of them and is denoted by A A A. An infinite union of a sequence of sets A,n is also defined as the set that consists of all. elementary events that belong to any one of them, An intersection of two events is an event whose elements belong to both of the events For two. events A and B their intersection is denoted by A B For a more general finite collection of. events A1 A2 An their intersection consists of all elementary events that belong to all of them. and is denoted by A A A, i 1 2 An An infinite intersection of a sequence of sets A.
is also defined as the set that consists of all elementary events that belong to all of them. Two events are called mutually exclusive if they cannot happen at the same time In the language. of set theory this simply means that they are disjoint sets i e sets that do not have any elements. in common or A B A finite collection of events A1 A2 An is said to be mutually. exclusive if any two events from the collection are mutually exclusive The concept is defined the. same way for infinite collections of events, ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2012 by Krzysztof Ostaszewski 4. GENERAL PROBABILITY, We say that a collection of events forms exhaustive outcomes or that this collection forms a. partition of the probability space if their union is the entire probability space and they are. mutually exclusive, An event A is a subevent although most commonly we use the set theoretic concept of a subset. of an event B denoted by A B if every elementary event sample point in A is also contained. in B This relationship is a mathematical expression of a situation when occurrence of A. automatically implies that B also occurs Note that if A B then A B B and A B A. For an event E its complement denoted by E C consists of all elementary events i e elements. of the sample space S that do not belong to E In other words E C S E where S is the entire. probability space Note that E E C S and E E C Recall that A B A BC is the set. difference operation, An important rule concerning complements of sets is expressed by DeMorgan s Laws. A B C AC BC A B C AC BC,A A1 A2 An C,A A1 A2 An C,An An An An.
n 1 n 1 n 1 n 1, The indicator function for an event E is a function I E S where S is the entire probability. space and is the set of real numbers defined as I E x 1 if x E and I E x 0 if x E. The simplest and commonly used example of a probability space is a set consisting of two. elements S 0 1 with 1 corresponding to success and 0 representing failure An. experiment in which only such two outcomes are possible is called a Bernoulli Trial You can. view taking an actuarial examination as an example of a Bernoulli Trial Tossing a coin is also an. example of a Bernoulli trial with two possible outcomes heads or tails you get to decide which. of these two would be termed success and which one is a failure. Another commonly used finite probability space consists of all outcomes of tossing a fair six. faced die The sample space is S 1 2 3 4 5 6 each number being a sample point representing. the number of spots that can turn up when the die is tossed The outcomes 1 and 6 for example. are mutually exclusive more formally these outcomes are events 1 and 6 The outcomes. 1 2 3 4 5 6 usually written just as 1 2 3 4 5 6 are exhaustive for this. probability space and form a partition of it The set 1 3 5 represents the event of obtaining an. ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2012 by Krzysztof Ostaszewski 5. odd number when tossing a die Consider the following events in this die tossing sample space. A 1 2 3 4 a number less than 5 is tossed,B 2 4 6 an even number is tossed. C 1 a 1 is tossed,D 5 a 5 is tossed,Then we have,A B 1 2 3 4 6 DC. A D i e events A and D are mutually exclusive,B C C 1 2 4 6 C 3 5 1 3 5 2 3 4 5 6 BC C C. Another rule concerning operations on events,A E1 E2 En A E1 A E2 A En.
A E1 E2 En A E1 A E2 A En, so that the distributive property holds the same way for unions as for intersections In particular. if E1 E2 En form a partition of S then for any event A. A A S A E1 E2 En A E1 A E2 A En, As the events A E1 A E2 A En are also mutually exclusive they form a partition of the. event A In the special case when n 2 E1 B E2 BC we see that A B and A BC form a. partition of A, Probability we will denote it by Pr is a function that assigns a number between 0 and 1 to each. event with the following defining properties, If En n 1 is a sequence of mutually exclusive events then Pr En Pr En. While the last condition is stated for infinite unions and a sum of a series it applies equally to. finite unions of mutually exclusive events and the finite sum of their probabilities. A discrete probability space or discrete sample space is a probability space with a countable. finite or infinite number of sample points The assignment of probability to each elementary. event in a discrete probability space is called the probability function sometimes also called. probability mass function, For the simplest such space described as Bernoulli Trial probability is defined by giving the.
probability of success usually denoted by p The probability of failure denoted by q is then. equal to q 1 p Tossing a fair coin is a Bernoulli Trial with p 0 5 Taking an actuarial. examination is a Bernoulli Trial and we are trying to get your p to be as close to 1 as possible. ASM Study Manual for Course P 1 Actuarial Examination Copyright 2004 2012 by Krzysztof Ostaszewski 6. GENERAL PROBABILITY, When a Bernoulli Trial is performed until a success occurs and we count the total number of. attempts the resulting probability space is discrete but infinite Suppose for example that a fair. coin is tossed until the first head appears The toss number of the first head can be any positive. integer and thus the probability space is infinite Repeatedly taking an actuarial examination. creates the same kind of discrete yet infinite probability space Of course we hope that after. reading this manual your probability space will not only be finite but a degenerate one. consisting of one element only, Tossing an ordinary die is an experiment with a finite probability space 1 2 3 4 5 6 If we. assign to each outcome the same probability of we obtain an example of what is called a. uniform probability function In general for a finite discrete prob. NO RETURN IF OPENED Study Manual for Exam P Exam 1 Probability 16 th Edition by Dr Krzysztof Ostaszewski FSA CERA FSAS CFA MAAA Note NO RETURN IF

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