ACTSC232-Ch2

ACTSC232-Ch2 - Chapter 2 Survival models Chapter 2 Survival...

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Unformatted text preview: Chapter 2 Survival models Chapter 2 Survival models ACTSC 232 Introduction to Actuarial Mathematics Tianxiang Shi Department of Statistics and Actuarial Science University of Waterloo Winter 2012 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Outline 1 Brief review of Prob. & Stat 2 The future lifetime random variable 3 The force of mortality 4 Moments of future lifetime Tx 5 Curtate future lifetime Kx Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Random variable Definition A random variable (r.v.) is a mapping from a set of random events to the real line R. discrete r.v. take values in a countable set (e.g. {0, 1}). continuous r.v. take values in some intervals of real numbers (e.g. [0, 1]). Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F (x) ≡ Pr(X ≤ x) for x ∈ R. A function F (x) to be a proper c.d.f. iff : Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F (x) ≡ Pr(X ≤ x) for x ∈ R. A function F (x) to be a proper c.d.f. iff : F (x) is non-decreasing as a function of x Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F (x) ≡ Pr(X ≤ x) for x ∈ R. A function F (x) to be a proper c.d.f. iff : F (x) is non-decreasing as a function of x lim F (x) = 0 x→−∞ Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Cumulative distribution function Definition A cumulative distribution function (c.d.f.) of a r.v. X is defined as F (x) ≡ Pr(X ≤ x) for x ∈ R. A function F (x) to be a proper c.d.f. iff : F (x) is non-decreasing as a function of x lim F (x) = 0 x→−∞ lim F (x) = 1 x→∞ Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Survival function Definition A survival function (s.f.) of a r.v. X is defined as S(x) ≡ Pr(X > x) for x ∈ R. A function S(x) to be a proper s.f. iff : Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Survival function Definition A survival function (s.f.) of a r.v. X is defined as S(x) ≡ Pr(X > x) for x ∈ R. A function S(x) to be a proper s.f. iff : S(x) is non-increasing as a function of x lim S(x) = 1 x→−∞ lim S(x) = 0 x→∞ Relationship : S(x) = 1 − F (x) Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Probability density/mass function Probability density function (p.d.f.) for a continuous r.v. X: f (x) = F (b) = d dx F (x) for x b f (x)dx −∞ ∈R Pr(a < X ≤ b) = Pr(a ≤ X < b) = b a f (x)dx Probability mass function (p.m.f.) for a discrete r.v. X: f (k) = Pr(X = k) for possible value k of X F (b) = k≤b f (k) Pr(X ≤ b) = Pr(X < b) is generally NOT TRUE! Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Moments of r.v.’s Definition The kth moment of a r.v. X is defined as E[X k ] = ∞ k −∞ x f (x)dx xk f (x) if X is continuous if X is discrete expectation/mean of X: E[X] variance of X: V ar(X) ≡ E (X − E[X])2 = E[X 2 ] − (E[X])2 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Some properties For some constants a, b, and r.v.s X, Y E[aX + bY ] = aE[X] + bE[Y ] For X and Y independent or uncorrelated V ar(aX + bY ) = a2 V ar(X) + b2 V ar(Y ) For a non-negative r.v. X ∞ E[X] = S(t)dt 0 and ∞ E[X 2 ] = 2 tS(t)dt, 0 provided that they exist. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Brief review of Prob. & Stat Conditional probability Definition The conditional probability of an event A, given an event B is given by Pr(AB) , Pr(A|B) = Pr(B) provided that Pr(B) = 0 Bayes formula Pr(A|B) = Tianxiang Shi(tim.shi@uwaterloo.ca) Pr(B|A) Pr(A) Pr(B) Chapter 2 Survival models The future lifetime random variable Future lifetime r.v. Notation Denote Tx as the future lifetime of (x) (a life aged x) for x ≥ 0. Tx is a non-negative continuous r.v. T0 is the future lifetime of a newborn and often with a limiting age ω (0 ≤ T0 ≤ ω) Age-at-death of (x) is Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Future lifetime r.v. Notation Denote Tx as the future lifetime of (x) (a life aged x) for x ≥ 0. Tx is a non-negative continuous r.v. T0 is the future lifetime of a newborn and often with a limiting age ω (0 ≤ T0 ≤ ω) Age-at-death of (x) is x + Tx Age-at-death of (x) is the death age of a newborn (0) conditional on (0)’s survival to age x: x + Tx = T0 |T0 ≥ x equivalently, Tx = T0 − x|T0 ≥ x Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Distribution of Tx Definition The probability that (x) does not survive beyond age x + t: t t qx := Pr(Tx ≤ t) = Fx (t) = fx (s)ds, 0 where fx (t) is the p.d.f. of Tx . Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Distribution of Tx Definition The probability that (x) does not survive beyond age x + t: t t qx := Pr(Tx ≤ t) = Fx (t) = fx (s)ds, 0 where fx (t) is the p.d.f. of Tx . Definition The probability that (x) survives to at least age x + t: ∞ t px := Pr(Tx > t) = Sx (t) = fx (s)ds. t Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Relationship between ages From Pr(Tx ≤ t) = Pr(T0 ≤ x + t|T0 > x), Fx (t) = S0 (x) − S0 (x + t) F0 (x + t) − F0 (x) = S0 (x) 1 − F0 (x) (we may also use S(t) and F (t) for a newborn). For the p.d.f. of Tx Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Relationship between ages From Pr(Tx ≤ t) = Pr(T0 ≤ x + t|T0 > x), Fx (t) = S0 (x) − S0 (x + t) F0 (x + t) − F0 (x) = S0 (x) 1 − F0 (x) (we may also use S(t) and F (t) for a newborn). For the p.d.f. of Tx f0 (x + t) = S0 (x) · fx (t) = x p0 · fx (t) For the s.f. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Relationship between ages From Pr(Tx ≤ t) = Pr(T0 ≤ x + t|T0 > x), Fx (t) = S0 (x) − S0 (x + t) F0 (x + t) − F0 (x) = S0 (x) 1 − F0 (x) (we may also use S(t) and F (t) for a newborn). For the p.d.f. of Tx f0 (x + t) = S0 (x) · fx (t) = x p0 · fx (t) For the s.f. Sx (t) = Tianxiang Shi(tim.shi@uwaterloo.ca) S0 (x + t) Sx (u + t) , and Sx+u (t) = S0 (x) Sx (u) Chapter 2 Survival models The future lifetime random variable Actuarial notation t = 1: px = 1 px and qx = 1 qx t px + t qx = 1 For s.f. of different ages x+t p0 = x p0 · t px = t p0 · x pt and in general u+t px Tianxiang Shi(tim.shi@uwaterloo.ca) = u px · t px+u = t px · u px+t Chapter 2 Survival models The future lifetime random variable Deferred mortality probability Definition Deferred mortality probability: the probability that (x) survives u years and then dies in the subsequent t years. That is, (x) dies between ages x + u and x + u + t, i.e. u+t u|t qx := Pr(u < Tx ≤ u + t) = fx (s)ds, u Remark: u|t qx = u px − u+t px = u+t qx − u qx u|t qx = u px · t qx+u Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Example Example Given that S(t) = e−0.01t , t ≥ 0 1 Verify that S(t) is a proper s.f. 2 Explain the meanings of 3 Explain the meaning of Tianxiang Shi(tim.shi@uwaterloo.ca) 15 p10 , 15 q10 5|15 q10 and calculate their values and calculate its value Chapter 2 Survival models The future lifetime random variable Relationships between ages cont’d For n ≥ 2, n px can be expressed as n px n qx = px px+1 · · · px+n−1 in terms of deferred mortality prob. n qx = m qx + m|n−m qx = m qx + m px · n−m qx+m for m < n (note that it holds for all m, n ∈ R+ ) and n qx = qx + 1| qx + 2| qx + · · · + n−1| qx If ω is the limiting age, then for n ≥ ω − x n qx = 1 = qx + 1| qx + 2| qx + · · · + n−1| qx Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The future lifetime random variable Example Example Given that 1| qx+1 = 0.095, 2| qx+1 = 0.171 and qx+3 = 0.2. Calculate qx+1 + qx+2 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Force of mortality Definition The force of mortality at age x is defined as µx := lim ∆x→0+ 1 Pr( T0 ≤ x + ∆x| T0 > x) ∆x for very small ∆x, we have Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Force of mortality Definition The force of mortality at age x is defined as µx := lim ∆x→0+ 1 Pr( T0 ≤ x + ∆x| T0 > x) ∆x for very small ∆x, we have µx · ∆x ≈ Pr( T0 ≤ x + ∆x| T0 > x) equivalently Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Force of mortality Definition The force of mortality at age x is defined as µx := lim ∆x→0+ 1 Pr( T0 ≤ x + ∆x| T0 > x) ∆x for very small ∆x, we have µx · ∆x ≈ Pr( T0 ≤ x + ∆x| T0 > x) equivalently µx = lim ∆x→0+ explicit expression Tianxiang Shi(tim.shi@uwaterloo.ca) 1 Pr(Tx ≤ ∆x) ∆x Chapter 2 Survival models The force of mortality Force of mortality Definition The force of mortality at age x is defined as µx := lim ∆x→0+ 1 Pr( T0 ≤ x + ∆x| T0 > x) ∆x for very small ∆x, we have µx · ∆x ≈ Pr( T0 ≤ x + ∆x| T0 > x) equivalently µx = lim ∆x→0+ 1 Pr(Tx ≤ ∆x) ∆x explicit expression −1 d d f0 (x) µx = S0 (x) = − ln S0 (x) = S0 (x) dx dx S0 (x) Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Some explanations The force of mortality is the sensitivity of death probability change to the change of age (given that the life will survive until x). Intuitively, it is the ”speed” of death at age x. Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Some explanations The force of mortality is the sensitivity of death probability change to the change of age (given that the life will survive until x). Intuitively, it is the ”speed” of death at age x. The force of mortality gives the conditional p.d.f. of age-at-death of a newborn at exact age x, given that the newborn’s survival to age x A typical graph of human mortality rates (see, textbook p.51, Figure 3.1) Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality General definition Definition µx (t) := lim ∆t→0+ Tianxiang Shi(tim.shi@uwaterloo.ca) 1 Pr( Tx ≤ t + ∆t| Tx > t) ∆t Chapter 2 Survival models The force of mortality General definition Definition µx (t) := lim ∆t→0+ 1 Pr( Tx ≤ t + ∆t| Tx > t) ∆t Aggregate mortality law: The force of mortality depends only on the attained age. µx (t) = µ0 (x + t) = µx+t In general, µx (t) = d fx (t) −1 d Sx (t) = − ln Sx (t) = Sx (t) dt dt Sx (t) µx is also called, failure/hazard rate Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Describe the future lifetime r.v. by µx s.f. of Tx Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Describe the future lifetime r.v. by µx s.f. of Tx t px = Sx (t) = e− p.d.f. of Tx Tianxiang Shi(tim.shi@uwaterloo.ca) x+t x µr dr = e− t 0 µx+s ds Chapter 2 Survival models The force of mortality Describe the future lifetime r.v. by µx s.f. of Tx t px = Sx (t) = e− x+t x µr dr = e− t 0 µx+s ds p.d.f. of Tx fx (t) = t px · µx+t = t px · µx (t) c.d.f. of Tx Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Describe the future lifetime r.v. by µx s.f. of Tx t px = Sx (t) = e− x+t x µr dr = e− t 0 µx+s ds p.d.f. of Tx fx (t) = t px · µx+t = t px · µx (t) c.d.f. of Tx t qx = 1 − e− Tianxiang Shi(tim.shi@uwaterloo.ca) x+t x µr dr = 1 − e− t 0 µx+s ds Chapter 2 Survival models The force of mortality Example Example 2 Given that S0 (x) = e−0.001x , x ≥ 0. Calculate 1 µx 2 f0 (t) 3 fx (t) Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Example Example 2 Given that S0 (x) = e−0.001x , x ≥ 0. Calculate 1 µx 2 f0 (t) 3 fx (t) Example (SOA Nov. 2000)Given that µx = F + e2x , x > 0 and Calculate F Tianxiang Shi(tim.shi@uwaterloo.ca) 0.4 p0 = 0.5. Chapter 2 Survival models The force of mortality Example cont’d Example (SOA Fall 2002)You are given (i) R = 1 − e− (ii) S = 1 − e− 1 0 µx (t)dt 1 0 (µx (t)+K)dt (iii) K is a constant such that S = 0.75R Find K in terms of qx or px Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Analytical laws of mortality De Moivre’s law (1729) µx = Tianxiang Shi(tim.shi@uwaterloo.ca) 1 ,0 ≤ x < ω ω−x Chapter 2 Survival models The force of mortality Analytical laws of mortality De Moivre’s law (1729) µx = 1 ,0 ≤ x < ω ω−x Gompertz’s law (1825) µx = Bcx , B > 0, c > 1, x ≥ 0 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Analytical laws of mortality De Moivre’s law (1729) µx = 1 ,0 ≤ x < ω ω−x Gompertz’s law (1825) µx = Bcx , B > 0, c > 1, x ≥ 0 Makeham’s law (1860) µx = A + Bcx , B > 0, A ≥ −B, c > 1, x ≥ 0 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models The force of mortality Analytical laws of mortality De Moivre’s law (1729) µx = 1 ,0 ≤ x < ω ω−x Gompertz’s law (1825) µx = Bcx , B > 0, c > 1, x ≥ 0 Makeham’s law (1860) µx = A + Bcx , B > 0, A ≥ −B, c > 1, x ≥ 0 Weibull’s law (1939) µx = kxn , k > 0, n > 0, x ≥ 0 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx The mean of future lifetime Tx Complete expectation of life The expected future lifetime of (x) is denoted as ∞ ˚x := E[Tx ] = e tfx (t)dt, 0 equivalently, Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx The mean of future lifetime Tx Complete expectation of life The expected future lifetime of (x) is denoted as ∞ ˚x := E[Tx ] = e tfx (t)dt, 0 equivalently, ∞ t · t px · µx+t dt ˚x = e 0 Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx The mean of future lifetime Tx Complete expectation of life The expected future lifetime of (x) is denoted as ∞ ˚x := E[Tx ] = e tfx (t)dt, 0 equivalently, ∞ ∞ t · t px · µx+t dt = ˚x = e 0 If there’s a limiting age ω − x, Tianxiang Shi(tim.shi@uwaterloo.ca) t px dt 0 Chapter 2 Survival models Moments of future lifetime Tx The mean of future lifetime Tx Complete expectation of life The expected future lifetime of (x) is denoted as ∞ ˚x := E[Tx ] = e tfx (t)dt, 0 equivalently, ∞ ∞ t · t px · µx+t dt = ˚x = e 0 t px dt 0 If there’s a limiting age ω − x, ω−x ˚x = e 0 Tianxiang Shi(tim.shi@uwaterloo.ca) ω−x t · t px · µx+t dt t px dt = 0 Chapter 2 Survival models Moments of future lifetime Tx Variance of the future lifetime Tx 2nd moment of Tx ∞ 2 E Tx = ∞ t2 fx (t)dt = t2 · t px · µx+t dt 0 0 equivalently, ∞ 2 E Tx = 2 t · t px dt 0 If there’s a limiting age ω − x, ω−x 2 E Tx = 2 t · t px dt 0 Variance of Tx 2 V ar(Tx ) = E Tx − [˚x ]2 e Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx Example Example (Constant force of mortality) Given that µx = µ, x ≥ 0. Calculate ˚x and V ar(Tx ) e Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx Example Example (Constant force of mortality) Given that µx = µ, x ≥ 0. Calculate ˚x and V ar(Tx ) e Example Given that µx = 1 ω−x , Tianxiang Shi(tim.shi@uwaterloo.ca) 0 ≤ x < ω. Calculate ˚x and V ar(Tx ) e Chapter 2 Survival models Moments of future lifetime Tx n-year temporary complete life expectancy of (x) Definition The n-year temporary complete life expectancy of (x) is the expectation of r.v. Tx ∧ n := min{Tx , n} = i.e., Tx , if Tx < n n, if Tx ≥ n ∞ n ˚x:n := E [Tx ∧ n] = e tfx (t)dt + 0 Integration by parts, Tianxiang Shi(tim.shi@uwaterloo.ca) nfx (t)dt n Chapter 2 Survival models Moments of future lifetime Tx n-year temporary complete life expectancy of (x) Definition The n-year temporary complete life expectancy of (x) is the expectation of r.v. Tx ∧ n := min{Tx , n} = i.e., Tx , if Tx < n n, if Tx ≥ n ∞ n ˚x:n := E [Tx ∧ n] = e tfx (t)dt + 0 nfx (t)dt n Integration by parts, n 0 Tianxiang Shi(tim.shi@uwaterloo.ca) n tfx (t)dt + n · n px = ˚x:n = e t px dt 0 Chapter 2 Survival models Moments of future lifetime Tx Recursive formula 2nd moment of Tx ∧ n n E (Tx ∧ n)2 t2 fx (t)dt + n2 · n px = 0 n t · t px dt = 2 0 Recursive formula ˚x = ˚x:n + n px˚x+n e e e In particular, e ˚x = ˚x:1 + px˚x+1 e e Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Moments of future lifetime Tx Example Example (SOA Nov. 2001) You are given that µx = Calculate ˚25:25 e Tianxiang Shi(tim.shi@uwaterloo.ca) 0.04, 0.05, x < 40 x ≥ 40 Chapter 2 Survival models Curtate future lifetime Kx Curtate future lifetime r.v. Definition The curtate future lifetime of (x) is the integer part of future lifetime, i.e. Kx := Tx For k = 0, 1, 2, . . . Pr(Kx = k) = Pr(k ≤ Tx < k + 1) Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Curtate future lifetime Kx Curtate future lifetime r.v. Definition The curtate future lifetime of (x) is the integer part of future lifetime, i.e. Kx := Tx For k = 0, 1, 2, . . . Pr(Kx = k) = Pr(k ≤ Tx < k + 1) = k| qx = k px − k+1 px and Pr(Kx ≤ k) = Pr(Tx < k + 1) = k+1 qx Why to analyze Kx ? Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Curtate future lifetime Kx Moments of Kx Mean of Kx ∞ k · k| qx = ex := E [Kx ] = k=0 Tianxiang Shi(tim.shi@uwaterloo.ca) ∞ k px k=1 Chapter 2 Survival models Curtate future lifetime Kx Moments of Kx Mean of Kx ∞ ∞ k · k| qx = ex := E [Kx ] = k=0 k px k=1 2nd moment of Kx ∞ ∞ E 2 Kx 2 k · k| qx = = k=0 (2k − 1)k px k=1 If there’s a limiting age ω, replace ∞ with Tianxiang Shi(tim.shi@uwaterloo.ca) Chapter 2 Survival models Curtate future lifetime Kx Moments of Kx Mean of Kx ∞ ∞ k · k| qx = ex := E [Kx ] = k=0 k px k=1 2nd moment of Kx ∞ ∞ E 2 Kx 2 k · k| qx = = k=0 (2k − 1)k px k=1 If there’s a limiting age ω, replace ∞ with ω − x − 1 ˚x (≥ ex ) ≈ ex + e Tianxiang Shi(tim.shi@uwaterloo.ca) 1 2 Chapter 2 Survival models Curtate future lifetime Kx n-year temporary curtate life expectancy of (x) n-year temporary curtate life expectancy The n-year temporary curtate life expectancy is the mean of Kx ∧ n = min{Kx , n}, i.e. n−1 ex:n := E [Kx ∧ n] = k · k| qx + n · n px = k=0 Tianxiang Shi(tim.shi@uwaterloo.ca) n k px k=1 Chapter 2 Survival models Curtate future lifetime Kx n-year temporary curtate life expectancy of (x) n-year temporary curtate life expectancy The n-year temporary curtate life expectancy is the mean of Kx ∧ n = min{Kx , n}, i.e. n−1 ex:n := E [Kx ∧ n] = n k · k| qx + n · n px = k=0 k px k=1 2nd moment of Kx ∧ n n−1 E (Kx ∧ n)2 = n k 2 · k| qx + n2 · n px = k=0 Recursive formula Tianxiang Shi(tim.shi@uwaterloo.ca) (2k − 1)k px k=1 ex = ex:n + n px ex+n Chapter 2 Survival models Curtate future lifetime Kx Example Example You are given that qx+k = 0.1(k + 1), k = 0, 1, 2, · · · , 9 Calculate ex:3 Tianxiang Shi(tim.shi@uwaterloo.ca) ...
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This note was uploaded on 03/30/2012 for the course ACTSC 232 taught by Professor Matthewtill during the Winter '08 term at Waterloo.

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