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Deeper Understanding, Faster Calculation --Exam P Insights & Shortcuts 17th Edition
by Yufeng Guo For SOA Exam P/CAS Exam 1 Spring, 2011 Edition
This electronic book is intended for individual buyer use for the sole purpose of preparing for Exam P. This book can NOT be resold to others or shared with others. No part of this publication may be reproduced for resale or multiple copy distribution without the express written permission of the author.
©2010, 2011, 2012 By Yufeng Guo
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http://actuary88.com/ Table of Contents
Chapter 1
Exam-taking and study strategy ........................................ 5
Top Horse, Middle Horse, Weak Horse.......................................................................... 5 Truths about Exam P....................................................................................................... 6 Why good candidates fail................................................................................................ 8 Recommended study method........................................................................................ 10 CBT (computer-based testing) and its implications...................................................... 11
Chapter 2
Doing calculations 100% correct 100% of the time ....... 13
What calculators to use for Exam P.............................................................................. 13 Critical calculator tips ................................................................................................... 17 Comparison of 3 best calculators.................................................................................. 26
Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8
Set, sample space, probability models ............................. 27 Multiplication/addition rule, counting problems ........... 41 Probability laws and “whodunit”..................................... 48 Conditional Probability..................................................... 60 Bayes’ theorem and posterior probabilities .................... 64 Random variables .............................................................. 74
Discrete random variable vs. continuous random variable........................................... 76 Probability mass function ............................................................................................. 76 Cumulative probability function (CDF)........................................................................ 79 PDF and CDF for continuous random variables........................................................... 80 Properties of CDF ......................................................................................................... 81 Mean and variance of a random variable...................................................................... 83 Mean of a function ........................................................................................................ 85
Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25
Independence and variance .............................................. 87 Percentile, mean, median, mode, moment........................ 91 Find E ( X ) ,Var ( X ) , E ( X | Y ) ,Var ( X | Y ) ................................ 95 Bernoulli distribution ....................................................... 111 Binomial distribution........................................................ 112 Geometric distribution ..................................................... 121 Negative binomial ............................................................. 129 Hypergeometric distribution ........................................... 140 Uniform distribution ........................................................ 143 Exponential distribution .................................................. 146 Poisson distribution .......................................................... 169 Gamma distribution ......................................................... 173 Beta distribution ............................................................... 183 Weibull distribution.......................................................... 193 Pareto distribution............................................................ 200 Normal distribution .......................................................... 207 Lognormal distribution.................................................... 212
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Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 Chapter 31 Chapter 32 Chapter 33
Chi-square distribution.................................................... 221 Bivariate normal distribution.......................................... 226 Joint density and double integration .............................. 231 Marginal/conditional density........................................... 251 Transformation: CDF, PDF, and Jacobian Method ..... 262 Univariate & joint order statistics .................................. 279 Double expectation............................................................ 296 Moment generating function ........................................... 302
14 Key MGF formulas you must memorize ............................................................... 303
Chapter 34 Chapter 35 Chapter 36
Joint moment generating function .................................. 326 Markov’s inequality, Chebyshev inequality................... 333 Study Note “Risk and Insurance” explained ................. 346
Deductible, benefit limit ............................................................................................. 346 Coinsurance................................................................................................................. 351 The effect of inflation on loss and claim payment...................................................... 355 Mixture of distributions .............................................................................................. 358 Coefficient of variation ............................................................................................... 362 Normal approximation ................................................................................................ 365 Security loading .......................................................................................................... 374
Chapter 37 On becoming an actuary… .............................................. 375 Guo’s Mock Exam ............................................................................... 377 Solution to Guo’s Mock Exam....................................................... 388 Final tips on taking Exam P..................................................................... 423 About the author....................................................................................... 424 Value of this PDF study manual.............................................................. 425 User review of Mr. Guo’s P Manual ....................................................... 425 Bonus Problems......................................................................................... 426
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!""#$%%&'"(&)*++,'-./ /
!"#$%&'()( (
*+#,-%#./01(#02(3%425(3%'#%&15(
/
67$(87'3&9(:/22;&(87'3&9( 95% E ( S ) *, . S is approximately normal. P )+ S > 95% E ( S ) *, = 1 $ P )+ S ! 95% E ( S ) *,
) 95% E ( S ) $ E ( S ) * ) E (S )* P )+ S ! 95% E ( S ) *, = : ; < = : ; $5% < = : )+ $5% (16.01) *, -S -S , + , + = : ( $0.80 ) = 1 $ : ( 0.80 ) P )+ S > 95% E ( S ) *, = 1 $ P )+ S ! 95% E ( S ) *, = 1 $ )+1 $ : ( 0.8 ) *, = : ( 0.8 ) = 0.7881
Example 15
You are given information about a block of insurance policies:
Class
# of policies
Probability that each policy has a loss in a year
Loss amount if there's a loss
1
100
0.1
3
2
200
0.2
4
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Let S represent the total annual loss incurred by the 300 policies.
-S
•
Calculate
•
Use normal approximation to calculate the probability that the total annual loss exceeds 105% of the expected total annual loss, that is P )+ S > 105% E ( S ) *,
E (S )
, the coefficient of variation of S .
Solution Let X represent the annual loss incurred by a policy randomly chosen from Class 1. Let Y represent the annual loss incurred by a policy randomly chosen from Class 2. S = X 1 + X 2 + ... + X 100 + Y1 + Y2 + ... + Y200 .
/ E ( S ) = 100 E ( X ) + 200 E (Y ) / Var ( S ) = 100Var ( X ) + 200Var ( Y ) "0 with probability of 0.9 X =% '3 with probability of 0.1 "0 Y =% '4
with probability of 0.8 with probability of 0.2
E ( X ) = 0 ( 0.9 ) + 3 ( 0.1) = 0.3
E ( X 2 ) = 02 ( 0.9 ) + 32 ( 0.1) = 0.9
Var ( X ) = E ( X 2 ) $ E 2 ( X ) = 0.9 $ 0.32 = 0.81
E (Y ) = 0 ( 0.8) + 4 ( 0.2 ) = 0.8
E (Y 2 ) = 0 2 ( 0.8 ) + 4 2 ( 0.2 ) = 3.2
Var ( Y ) = E (Y 2 ) $ E 2 (Y ) = 3.2 $ 0.82 = 2.56
To avoid the above calculation, you can use BA II Plus and quickly calculate the mean and variance for X and Y . / E ( S ) = 100 E ( X ) + 200 E ( Y ) = 100 ( 0.3) + 200 ( 0.8 ) = 190 / Var ( S ) = 100Var ( X ) + 200Var (Y ) = 100 ( 0.81) + 200 ( 2.56 ) = 593
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/
-S
E (S )
=
593 = 0.1282 190
Next, we need to find P )+ S > 105% E ( S ) *, . S is approximately normal. P )+ S > 105% E ( S ) *, = 1 $ P )+ S ! 105% E ( S ) *,
)105% E ( S ) $ E ( S ) * ) E (S )* P )+ S ! 105% E ( S ) *, = : ; < = : ;5% < = : )+5% ( 7.8 ) *, -S -S , + , + = : ( 0.39 ) = 0.6517 P )+ S > 105% E ( S ) *, = 1 $ P )+ S ! 105% E ( S ) *, = 1 $ 0.6517 = 0.3483
Example 16
Two random loss variables, X and Y have the following joint density function: f X ,Y ( x, y ) =
4 xy, 81
where 0 ! X ! 3 and 0 ! Y ! 3
The insurer pays X + Y in excess of deductible 1, subject to the maximum payment of 2. Calculate the expected claim payment by the insurer. Solution
Let S represent the claim payment by the insurer. Then, "0 # S = % X + Y $1 #2 '
if 0 ! X + Y ! 1 if 1 ! X + Y ! 3 if 3 ! X + Y
Next, we draw a diagram
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In the above diagram, the square ODEC is where X and Y exist. AB represents x + y = 1 . CD represents x + y = 3 . Area AOB is where 0 ! x + y ! 1 . When ( x, y ) falls in AOB, the insurer pays nothing. Area ABDC is where 1 ! x + y ! 3 ; here the insurer pays x + y $ 1 . Area CDE is where 3 ! x + y , x ! 3 , and y ! 3 ; here the insurer pays 2.
"0 # S = % X + Y $1 #2 ' E (S ) =
33
if 0 ! X + Y ! 1 if 1 ! X + Y ! 3 if 3 ! X + Y
0 f ( x, y ) dxdy +
AO B
33
33 ( x + y $ 1) f ( x, y ) dxdy + 33
A BDC
2 f ( x, y ) dxdy
CDE
0 f ( x, y ) dxdy = 0
AO B
4 33 ( x + y $ 1) f ( x, y ) dxdy = 33 ( x + y $ 1) 81 x y dx dy .
A BDC
A B DC
To do this integration, we divide ABDC into two areas: ABDF and AFC.
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4 4 4 33 ( x + y $ 1) 81 x y dx dy = 33 ( x + y $ 1) 81 x y dx dy + 33 ( x + y $ 1) 81 x y dx dy
A BDC
A BDF
A FC
1 3$ x 1 ) * 4 4 56 8 20 ( x + y $ 1) x y dx dy = 3 ; 3 ( x + y $ 1) x ydy < dx = 3 46 x $ x 2 57 dx = 33 81 81 243 81 9 243 A BDF 0 +1$ x 08 , 3 3$ x ) * 4 4 184 ( x + y $ 1) x y dx dy = 3 ; 3 ( x + y $ 1) x ydy < dx = 33 81 81 1215 A FC 1 + 0 ,
20 184 33 ( x + y $ 1) f ( x, y ) dxdy = 243 + 1215 = 0.2337
A BDC
3 )3 44 5 * 5 2 f x , y dxdy = ( ) ; 3 2 6 xy 7 dy K 95 0.5, K 95 0.5@ and K d K 95 becomes K d K 95 0.5 . (You can ignore the continuity correction factor in this problem because 0.5/
K 95
0.5 20 1.645 19.6
happens to be small.)
27.783
X 95 100 K 95 =100*27.783=2,778
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http://actuary88.com Q29 Random variables X and Y have the following joint pdf: f x, y k
x , where 0 x y 1 and k is a constant. y
§ Calculate E ¨ X Y ©
2· § ¸ Va r¨ X Y 3¹ ©
A 0.17
B 0.27
Solution
D
§ f ¨x y ©
C 0.37
2· § f ¨ x, y ¸ 3¹ © 2· § P¨ y ¸ 3¹ ©
2· ¸ 3¹
2· ¸ 3¹
2 3
D 0.47
3 kx 2 §
2· ¸ dx 3¹
³ f ¨© x, y 0
2 where 0 x and 3
3 k 2
§ P¨ y ©
2· ¸ 3¹
2 3
2 3
3 kx 2
§
0
3
0
c is a constant. 2· ¸ dx 3¹
§ You don’t need to worry about actually finding k or P ¨ y © 2· ¸ 3¹
§ f ¨x y ©
2 3
§
§ f ¨x y ©
2· ¸ . You just need to find c . 3¹
cx should add up to one over the range 0 x
³ f ¨© x y 0
cx ,
³ 2 kxdx
3 k 2
³ f ¨© x, y
E 0.57
2· ¸dx 3¹ 2· ¸ 3¹
2 : 3
2 3
³ cxdx
1, c
4.5 .
0
4.5 x , where 0 x
2 . 3
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§ So E ¨ X Y © ª§ E «¨ X Y ¬«©
2· º ¸ » 3 ¹ ¼»
§ Var ¨ X Y ©
2· ¸ 3¹
§ E¨ X Y ©
2 3
2· ¸ 3¹
2· ¸dx 3¹
§ ³0 xf ¨© x y 2 3
2
§ ³0 x f ¨© x y 2
2· ¸dx 3¹
2 3
³ x 4.5 x dx
2 3 3 0
>1.5x @
0
2 3
§2· 1.5¨ ¸ ©3¹ 2
ª 4.5 4 º 3 « 4 x » ¬ ¼0
³ x 4.5 x dx 2
0
3
4 =0.444 9
4.5 § 2 · ¨ ¸ 4 ©3¹
4
2 9
2
2 §4· ¨ ¸ =0.02469 9 ©9¹
2· ¸ Va 3¹
§ r¨ X Y ©
2· ¸ =0.4444 + 0.02469=0.47 3¹
Q30 1 , 1Tt where T is a positive constant. Calculate the probability that X is one standard deviation from its mean.
Random variable X has the following moment generating function: M X t
A 0.46
B 0.56
Solution
E
C 0.66
D 0.76
E 0.86
Notice that X is an exponential random variable with mean T and standard deviation of T. P ª¬ V X d X E X d V X º¼
1 e
2T
T
1 e 2
P T d X T d T
P 0 d X d 2T
F 2T
0.8647
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Final tips on taking Exam P 1. The goal of combat is to win the war, not individual battles. The goal of taking Exam P is to get a 6 so you can look for a job, not to get a 10. 2. If you need to speak in an important meeting (such as before Congress), always prepare a script well in advance and rehearse your script. When preparing for an important exam such P, always build a 3 minute solution script ahead of time for each of the tested problems in Sample P and any newly released P exams (if any). Then when taking the exam, just regurgitate the script and solve the repeatedly tested problems. 3. In the exam, if a problem is brand new (this type of problems have not been tested before), make one attempt. If you can’t solve, just guess an answer, and move on to the next problem. 4. Focus on mastering the fundamentals. Difficult distributions such as hypergeometric, Weibull, Pareto, beta, Chi-square, and bivariate normal aren’t tested in the Sample Exam P problems. Since Sample Exam P reflects the level of knowledge you need to have to pass Exam P, chances are that these difficult distributions won’t show up in your exam. As a result, you can probably ignore these distributions. However, if you really can’t sleep well if you ignore these distributions, you might learn some basic knowledge about these distributions. Don’t over-study these difficult distributions. 5. Master Sample Exam P before taking your exam. Put yourself in the exam-like condition and practiced Sample P exam and any newly released P exams (if any). Work and rework until you can solve Sample P exam and any newly released P exams (if any) 100% right in the exam-like condition. Never walk into the exam room without mastering Sample P exam and any newly released P exams (if any). 6. For those wanting for additional practice problems, refer to http://www.actuarialoutpost.com/actuarial_discussion_forum/. You’ll find many practice problems there. 7. If you failed Exam P, don’t give up. Many candidates eventually passed Exam P after failing it multiple times.
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About the author Yufeng Guo was born in central China. After receiving his Bachelor’s degree in physics at Zhengzhou University, he attended Beijing Law School and received his Masters of law. He was an attorney and law school lecturer in China before immigrating to the United States. He received his Masters of accounting at Indiana University. He has pursued a life actuarial career and passed exams 1, 2, 3, 4, 5, 6, and 7 in rapid succession after discovering a successful study strategy. Mr. Guo’s exam records are as follows: Fall 2002 Passed Course 1 Spring 2003 Passed Courses 2, 3 Fall 2003 Passed Course 4 Spring 2004 Passed Course 6 Fall 2004 Passed Course 5 Spring 2005 Passed Course 7 Mr. Guo currently teaches an online prep course for Exam P, FM, MFE, and MLC. For more information, visit http://actuary88.com. If you have any comments or suggestions, you can contact Mr. Guo at
[email protected].
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Value of this PDF study manual 1. Don’t pay the shipping fee (can cost $5 to $10 for U.S. shipping and over $30 for international shipping). Big saving for Canadian candidates and other international exam takers. 2. Don’t wait a week for the manual to arrive. You download the study manual instantly from the web and begin studying right away. 3. Load the PDF in your laptop. Study as you go. Or if you prefer a printed copy, you can print the manual yourself. 4. Use the study manual as flash cards. Click on bookmarks to choose a chapter and quiz yourself. 5. Search any topic by keywords. From the Adobe Acrobat reader toolbar, click Edit ->Search or Edit ->Find. Then type in a key word.
User review of Mr. Guo’s P Manual Mr. Guo’s P manual has been used extensively by many Exam P candidates. For user reviews of Mr. Guo’s P manual at http://www.actuarialoutpost.com, click here Review of the manual by Guo. Testimonies: “Second time I used the Guo manual and was able to do some of the similar questions in less than 25% of the time because of knowing the shortcut.” Testimony #1 of the manual by Guo “I just took the exam for the second time and feel confident that I passed. I used Guo the second time around. It was very helpful and gives a lot of shortcuts that I found very valuable. I thought the manual was kind of expensive for an e-file, but if it helped me pass it was well worth the cost.” Testimony # 2 of the manual by Guo “I took the last exam in Feb 2006, and I ran out of time and I ended up with a five. I needed to do the questions quicker and more efficiently. The Guo's study guide really did the job.” Testimony #3 of the manual by Guo
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Bonus Problems
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W
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# B $ @ '9& :( B$ " $ 8 '9& :( ) ) +9 : ! +9: $ & : ) ) * '9 ! :( B9B: B9B: %!%! %!%! ! , '$ ( ) % ( :8 '9& :( A9A: ) % ( *: '9 ! :( A9A: ) % ,4#".5-. 6(7 %! A@ ':( , '$ ( ) % :8 ':( A: 8 ':( ) A:
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