Actuarial Science 2427A/B Lecture Notes - Lecture 10: Kolmogorov Space, Survival Analysis, Reliability Engineering
AS2427a Chapter 2 Lecture Notes January 2018
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Lecture note handouts will have to be modified in class to properly reflect full notation
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These posted notes may be updated as we work through the material. Any changes or
addition to these notes will be announced
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AS2427a Chapter 2 Lecture Notes January 2018
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(īīīīīīīīīīī,īīīīīīīīīīīīīī-ī
(x)
ī
an individual life currently aged x
Ļ
ĻĻ
Ļ
ī
limiting age (age at which there are no survivors)
T
x
ī
continuous random variable(r.v) for future lifetime of (x)
and (x+T
x
) is the age-at death r.v. for (x)
F
x
ī
distribution function(cdf) for T
x
also referred to as the lifetime distribution for (x)
Fx(t) = Pr[Tx ī t] =
Pr[
(x) doesnāt survive beyond age
(x+t)]
S
x
ī
survival function
Sx(t) =1āFx(t) = Pr[ Tx > t
] = Pr[
(x) survives at least t years]
f
x
ī
the pdf for T
x
(
will use
fx(t)
in 2.3)
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AS2427a Chapter 2 Lecture Notes January 2018
3
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Require,
Pr [Tx ī t ] = Pr [T0 ī x+tīT0 > x] (1)
= Pr[x < T0 ī x+t ]/Pr[T0 >x]
So Fx(t)= Pr[ Tx ī t] = [F0(x+t) ā F0(x)]/S0(x)) (2)
And 1ā Fx(t) = Sx(t) = S0(x+t)/S0(x) (3)
Rearranging, S0(x+t) = S0(x)Sx(t) (4)
Sx(t+u) = S0(x+t+u)/S0(x) (mult by S0(x+t)/S0(x+t))
= [S0(x+t)/S0(x)][(S0(x+t+u)/S0(x+t)]
Sx
(t+u) = Sx
(t) Sx+t
(u) (5)
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