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Product Preview PUBLICATIONS e Experts In Actuarial Career Advancement Product Preview For More Information: email [email protected] or call 1(800) 282-2839 Chapter 1 – Actuarial Notation 1 CHAPTER 1 – ACTUARIAL NOTATION There is a specific notation used for expressing probabilities, called “actuarial notation”. You must become familiar with this notation, as it is used extensively in the study material and on the exams. The notation txp means “the probability of a person age x living t or more years”. Example 1.1 – Interpret 57p Answer – The probability of a person age 7 living five or more years. Conversely, txq means “the probability of a person age x dying within the next t years”. Example 1.2 – Interpret 57q Answer – The probability of a person age 7 dying at or before age 12. In the case where we are only looking for the probability over the course of one year we use the shorthand notation px or qx . Example 1.3 – Interpret q0 Answer – The probability of a person age 0 dying at or before age 1. Even though we are using a different notation we still follow the same rules of probability. One of the more common rules used is that if you have two (and only two) possible events, A and B, then PA() PB () 1. The probabilities of these two events are compliments of each other. Example 1.4 – Given 510p 0.9 , find 510q Answer – These two values are compliments of each other so we can say 510qp1 5 10 1 0.9 0.1 Because of the way we write the notation, it can be a little confusing to try to relate the different probabilities to each other. Example 1.5 – Given 23p 0.9 and 45p 0.8 , find 63p Answer – By rule, 63p 2345pp. This is true because in order for a person age 3 to live 6 or more years (i.e. 63p ) he must first live 2 more years (i.e. 23p ) and then live another 4 more years (i.e. 45p ). So in this case 63p 0.9 0.8 0.72 . Example 1.6 – Given 46q 0.2 and 26p 0.9 , find 28p => Answer – Let’s solve this problem in steps: p pp p I. Here 46 2628, so we first must find 46 © ACTEX 2015 CAS Exam LC Study Manual 2 Chapter 1 – Actuarial Notation p 1 q p 1 0.2 0.8 II. By the property of complements, 46 46, so 46 . 0.8 0.9 p p 0.8889 III. Then we can say 28, so 28 . Just because 20p pp 1 0 doesn’t mean that 20qqq 1 0. You must remember that 20q is the probability that a person age 0 dies within the first two years – the person could die in the first year or survive the first year and die in the second year. So we would write this as 20qqpq 0 0 1, where q0 = probability of dying in the first year p01q = probability of surviving the first year and dying in the second year By extension we can also say 30q q 0 pq 0 1 2 pq 0 2. Example 1.7 – Given 21p 0.6 and q1 0.1, find q2 Answer – We can solve this two ways: First way: I. 21p pp 1 2 II. pq111 1 0.1 0.9 0.6 0.9 p2 p2 0.6667 III. 21p pp 1 2 => => q 1 0.6667 0.3333 IV. 2 Second way: I. 21qqpq 1 1 2 II. 21qp1 2 1 1 0.6 0.4 III. pq111 1 0.1 0.9 IV. 0.4 0.1 0.9 q2 => q2 0.3333 The final type of notation is one that is used when we would like to “delay” the death or failure. For example, q means the probability of a person age 8 surviving for 3 years and then dying within the 3 48 next four years. The number to the left of the bar means that the person will survive at least that number of years before dying. We can calculate q using the equation qpq. Similarly, q is 3 48 3 48 38411 4 0 the probability of a person age 0 surviving for 4 years and dying within the next year ( qpq). 4 0404 Example 1.8 – Given p 0.9 and p 0.8 , find q . 20 22 2 20 Answer – I. qpq 2 20 2 0 22 II. 22qp1 2 2 1 0.8 0.2 . III. So q 0.9 0.2 0.18 . 2 20 © ACTEX 2015 CAS Exam LC Study Manual Chapter 1 – Actuarial Notation 3 Chapter 1 – Questions Express the following using actuarial notation: 1) The probability of a person age 60 dying within 2 years. 2) The probability of a person age 10 living one or more year. 3) The probability of a newborn living at least 20 years. 4) The probability of a person age 2 dying within half a year. 5) The probability of a person age 5 living 3 years and then dying before age 10. 6) The probability of a newborn living 20 years and then dying within the next year. Solve the following: 7) Given p0 0.8 , find q0 8) Given 22q 0.01, find 22p . 9) Given p1 0.8 and 32p 0.7 , find 41p 10) Given 73p 0.36 , p3 0.9 , and 24p 0.8 , find 46p 11) Given 24p 0.75 and q4 0.1, find p5 . 12) Given 31q 0.2 and q4 0.15 , find 41p . 13) Given 0.5p 2 0.7 and 1.5p 2.5 0.6 , find 22p . 14) Given p 0.8 and q 0.2 , find q 22 34 2 32 15) Given p 0.9 and p 0.7 , find q . 310 13 3 10 16) Given p 0.54 , p 0.5, and p 0.9 , find q 45 9 4 5 4 17) Given p 0.9 , p 0.7 , and p 0.6 , find q . 2 3 4 2 2 © ACTEX 2015 CAS Exam LC Study Manual 4 Chapter 1 – Actuarial Notation Chapter 1 – Answers Express the following using actuarial notation: 1) The probability of a person age 60 dying within 2 years. Answer: 260q 2) The probability of a person age 10 living one or more year. Answer: p10 3) The probability of a newborn living at least 20 years. Answer: 20p 0 4) The probability of a person age 2 dying within half a year. Answer: 0.5q 2 5) The probability of a person age 5 living 3 years and then dying before age 10. Answer: q 3 25 6) The probability of a newborn living 20 years and then dying within the next year. Answer: p 20 0 Solve the following: 7) Given p0 0.8 , find q0 Answer: qp001 1 0.8 0.2 8) Given 22q 0.01, find 22p . Answer: 22pq1 22 1 0.01 0.99 9) Given p1 0.8 and 32p 0.7 , find 41p Answer: 41ppp 132 0.8 0.7 0.56 10) Given 73p 0.36 , p3 0.9 , and 24p 0.8 , find 46p Answer: 73p ppp 32446 => 0.36 0.9 0.8 46p => 46p 0.5 © ACTEX 2015 CAS Exam LC Study Manual Chapter 1 – Actuarial Notation 5 11) Given 24p 0.75 and q4 0.1, find p5 . Answer: 24p pp 4 5 => 0.75 0.9 p5 => p5 0.8333 12) Given 31q 0.2 and q4 0.15 , find 41p . Answer: 41p 31pp 4 => 41p 0.8 0.85 0.68 13) Given 0.5p 2 0.7 and 1.5p 2.5 0.6 , find 22p . Answer: 2ppp 2 0.5 2 1.5 2.5 0.7 0.6 0.42 14) Given p 0.8 and q 0.2 , find q . 32 34 2 32 Answer: qpq0.8 0.2 0.16 2 32 2 2 34 15) Given p 0.9 and p 0.7 , find q . 310 13 3 10 Answer: qpq0.9 0.3 0.27 3 10 3 10 13 16) Given p 0.54 , p 0.5, and p 0.9 , find q 45 9 4 5 4 Answer: We need to solve this in a few steps: I. qpq 5 4549 II. 54p pp 445 => 54p 0.9 0.54 0.486 III. q 0.486 0.5 0.243 5 4 17) Given p 0.9 , p 0.7 , and p 0.6 , find q . 2 3 4 2 2 Answer: qpqppq0.9 0.7 0.4 0.252 2 2224 234 © ACTEX 2015 CAS Exam LC Study Manual 6 Chapter 1 – Actuarial Notation © ACTEX 2015 CAS Exam LC Study Manual Chapter 2 – Survival Models 7 CHAPTER 2 – SURVIVAL MODELS A survival model is a model for a continuous, random variable T0 , which will “fail” at some age x. Some example of survival models include: 1) Duration of a battery 2) Duration of a baseball strike 3) Lifetime of a person T0 = random variable for age of entity when it fails, assuming it starts at age 0. This would be the random variable used when modeling the survival of a person from birth. Tx = random variable for the time of survival for an entity, assuming that it is alive at age x. For example, if a person aged 30 decides to purchase life insurance the, random variable would be T30 and an actuary would model the years survived past age 30.
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