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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 nondecreasing 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 nondecreasing 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 nondecreasing 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 nonincreasing 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 nonnegative 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(AB) =
Pr(B)
provided that Pr(B) = 0
Bayes formula
Pr(AB) = Tianxiang Shi(tim.shi@uwaterloo.ca) Pr(BA) 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 nonnegative continuous r.v.
T0 is the future lifetime of a newborn and often with a
limiting age ω (0 ≤ T0 ≤ ω)
Ageatdeath 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 nonnegative continuous r.v.
T0 is the future lifetime of a newborn and often with a
limiting age ω (0 ≤ T0 ≤ ω)
Ageatdeath of (x) is x + Tx
Ageatdeath 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 − xT0 ≥ 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 + tT0 > 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 + tT0 > 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 + tT0 > 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
ut qx := Pr(u < Tx ≤ u + t) = fx (s)ds,
u Remark:
ut qx = u px − u+t px = u+t qx − u qx ut 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 515 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 + mn−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
ageatdeath 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 nyear temporary complete life expectancy of (x)
Deﬁnition
The nyear 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 nyear temporary complete life expectancy of (x)
Deﬁnition
The nyear 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 nyear temporary curtate life expectancy of (x)
nyear temporary curtate life expectancy
The nyear 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 nyear temporary curtate life expectancy of (x)
nyear temporary curtate life expectancy
The nyear 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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