Survival DIstributions

THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2801 Life Contingencies
LN1: Chapter 3 (Actuarial Mathematics): Survival distributions
Age-at-death random variable
T0 – age-at-death (lifetime for newborn) random variable
To completely determine the distribution of T0 , we may use (for t ≥ 0),
(1) (cumulative) distribution function: F0 (t) = Pr(T0 ≤ t)
(2) survival function: s0 (t) = 1 − F0 (t) = Pr(T0 > t)
(3) probability density function: f0 (t) = F0 (t) =
(4) force of mortality: µ0 (t) =

d
F0 (t) dt f0 (t)
−s0 (t)
=
1 − F0 (t) s0 (t)

Requirements:
(1) For distribution function,
F(x)
1

(i) non-decreasing function

x

0

(ii) F0 (t) → 1 as t → ∞
(iii) F0 (t) = 0,

1

t≤0

Relations:

• F0 (t) =

t
0

f0 (y)dy

• F0 (t) = 1 − s0 (t)
• F0 (t) = 1 − exp{−

t
0

µ0 (y)dy}

(2) For survival function, s(x) 1

(i) non-increasing function

x

0

(ii) s0 (t) = 1 as t ≤ 0
(iii) s0 (t) → 0,

t→∞

Relations:

• s0 (t) =

∞ t f0 (y)dy

• s0 (t) = 1 − F0 (t)
• s0 (t) = exp{−

t
0

µ0 (y)dy}

(3) For probability density function,

(i) f0 (t) = 0,

t t) = 1 − t qx ,

t ≥ 0.

If t = 1, we write 1 qx and 1 px as qx and px , respectively. Note that t px

= Pr(Tx > t) = Pr(T0 − x > t|T0 > x) =

t qx

=1−

s0 (x + t)
.
s0 (x)
3

s0 (x + t)
Pr((T0 > x + t) ∩ (T0 > x))
=
,
Pr(T0 > x) s0 (x)

The probability that (x) dies between ages x + t and x + t + u is given by

|

|

|

x

x+t

x+t+u

t|u qx

= Pr(t < Tx < t + u)
= Pr(Tx ≤ t + u) − Pr(Tx ≤ t)
=

t+u qx

− t qx

= (1 − t+u px ) − (1 − t px )

 t+u px =

= t px − t+u px ←
=


=
= t px − t px u px+t
=
=
If u = 1, we write

t px (1

s0 (x+t+u) s0 (x) s0 (x+t) s0 (x+t+u)
· s0 (x+t) s0 (x)

t px u px+t

t px u qx+t .

t|1 qx

− u px+t )

as t| qx

Example 1.
Suppose the survival function of the future lifetime of a newborn