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Actuarial Science

ACTSC 232

Jun Cai

Fall

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Summary of Lecture Notes - ACTSC 232, Winter 2010
Part 1 - Survival Distributions and Life Tables
1.1 Future Lifetime Random Variables
(a) Let (x) denote a life or person aged x( 0) at time 0 or at the current time.
(b) Let T denote the time-until-death of (x). That is to say that (x) will die at age
x
x+T or will die in T years or T is the death time of (x). The random variable T
x x x x
is called the future lifetime of a life (x). Note that Tx> 0 is a continuous random
variable.
(c) The distribution function (d.f.) of T is denoted by
x
Fx(t) = PrfT x tg; t 0:
The survival distribution (s.f.) of Txis denoted by
S xt) = PrfT >xtg = 1 F (t)x t 0:
The probability density function (p.d.f.) of T is denoted by
x
d d
fx(t) = Fx(t) = S xt); t 0:
dx dx
(d) Let T denote the age at the death of a newborn life or the future lifetime of (0), or
0
the age at the death of (x) or the future lifetime of (x) from his birth.
Note that T 0 0 is a continuous random variable. When we view T as the 0ge at
the death of (x) or the future lifetime of (x) from his birth, then T =0x + T andx
Txis as a conditional random variable conditioning on T > x.0In this course, we
explain T xn this way.
(e) The distribution function (d.f.) of T0is denoted by
F (x) = PrfT xg:
0 0
The survival distribution (s.f.) of T0is denoted by
S 0x) = 1 F (0) = PrfT > 0g:
The probability density function (p.d.f.) of X is denoted by
d d
f 0x) = F 0x) = S0(x):
dx dx
1 (f) Relationships between the distribution of x and the distribution of T0:
PrfT x tg = PrfT x 0 tjT > xg;0
PrfT x tg = PrfT > x 0 tjT > xg:0
That means
F 0x + t) F0(x) S0(x) S0(x + t)
F xt) = 1 F (x) = S (x)
0 0
S0(x + t)
S xt) = :
S0(x)
(g) Review of the properties of distribution and survival functions: Let the df and sf of
a r.v. Y be F(x) = PrfY xg and S(x) = PrfY > xg.
i. 0 F(x) 1 (0 S(x) 1).
ii. F(x) (S(x)) is a non-decreasing (non-increasing) and right-continuous function.
iii. lit!1 F(x) = 1 and lim t!1 F(x) = 0 (lim t!1 S(x) = 0 and lim t!1 S(x) =
1).
iv. For any a < b,
Prfa < Y bg = F(b) F(a) = S(a) S(b):
v. If F(x) is continuous, then for any a < b,
Prfa < Y bg = Prfa < Y < bg = Prfa Y bg = Prfa Y bg
= F(b) F(a) = S(a) S(b):
vi. If Y is limited or bounded from above, say Y !, then
F(x) = 1; x !
S(x) = 0; x !:
Note that if Y represents future lifetime, the ! is called the limiting age.
vii. The conditions (i)-(iii) are sucient and necessary for a function to be a distri-
bution (survival) function of a random variable.
1.2 The Force of Mortality
Let Y > 0 be a positive and continuous r.v. with df F(x) = PrfY xg, pdf f(x), and sf
S(x) = PrfY > xg. Dene function h(x) by
f(x) dS(x) d
h(x) = = dx = logS(x):
S(x) S(x) dx
2This function h(x) is called \hazard rate function" or \failure rate function" of Y in
statistics and probability and \force of mortality" of Y in actuarial mathematics.
Note that Y > 0 and S(0) = 1. For any x 0,
Z x Z x
h(t)dt = dlogS(t) = (logS(x) logS(0)) = logS(x):
0 0
R
xh(t)dt
Therefore, for any x 0, S(x) = e 0 :
For the positive and continuous random variable Y , the pdf f(x), df F(x), sf S(x), and
the force of mortality h(x) are equivalent in the sense that given any of the four functions,
we can determine any other three functions by the following relationships: for any x 0,
Rx
S(x) = e 0h(t)d;
f(x) d S(x) d
h(x) = = dx = logS(x);
S(x) S(x) dx
d d
f(x) = F(x) = S(x);
Zxx dx
F(x) = f(t)dt:
0
(a) We denote the force of mortality of T b0 anx

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