Exam M Notes F05

Chapter 3: Survival Distributions and Life Tables
Distribution function of X:

Force of mortality µ(x): fX (x)
1 − FX (x) s (x)
= − s(x) FX (x) = Pr(X ≤ x)

µ(x) =

Survival function s(x): s(x) = 1 − FX (x)

Relations between survival functions and force of mortality:
 x


Probability of death between age x and age y:
Pr(x < X ≤ z) = FX (z) − FX (x)

s(x) = exp −

= s(x) − s(z)

Pr(x < X ≤ z|X > x) =
=



µ(y)dy  x FX (z) − FX (x)
1 − FX (x) s(x) − s(z) s(x) Derivatives: d t qx dt d t px dt d
Tx
dt d Lx dt d
˚
ex dt = Pr[T (x) ≤ t]
= prob. (x) dies within t years
= distribution function of T (x)

t px

= exp −

n px

Notations: t qx

0 x+n 

Probability of death between age x and age y given survival to age x:

µ(y)dy 

= Pr[T (x) > t]
= prob. (x) attains age x + t

=

t px

· µ(x + t) = fT (x) (t)

= − t px · µ(x + t)
= −lx
= −dx
= µ(x)˚ ex − 1

Mean and variance of T and K:

= 1 − t qx

E[T (x)] ≡ complete expectation of life t|u qx

= Pr[t < T (x) ≤ t + u]
=

t+u qx

− t qx

=

t px

−

t+u px

=

t px

∞

≡ ˚ ex =
0

E[K(x)] ≡ curtate expectation of life

· u qx+t

∞

≡ ex =

Relations with survival functions: t px t qx

k px k=1 s(x + t) s(x) s(x + t)
= 1− s(x) ∞

=

V ar[T (x)] = 2

t · t px dt − ˚ e2x 0
∞
k=1

Pr[K(x) = k] = Pr[k ≤ T (x) < k + 1]
=

k px −

=

k px · qx+k

=

k| qx

(2k − 1) k px − e2x

V ar[K(x)] =

Curtate future lifetime (K(x) ≡ greatest integer in T (x)):

Exam M - Life Contingencies - LGD c

t px dt

Total lifetime after age x: Tx

k+1 px

∞

Tx =

lx+t dt
0

1

Total lifetime between age x and x + 1: Lx

Central death rate: mx

Lx = Tx − Tx+1
1

=

1

lx · t px dt

lx+t dt =
0

lx − lx+1
Lx
lx − lx+n n Lx

mx = n mx

=

0

Total lifetime from age x to x + n:

Fraction of year lived between age x and age x + 1 by dx : a(x)

n Lx

n−1 n Lx = Tx − Tx+n =

Lx+k

1

t · t px · µ(x + t) dt

k=0 n =

a(x) =

t px

Recursion formulas:
E[K] = ex = px (1 + ex+1 )

Tx lx E[T ] = ˚ ex = px (1 + ˚ ex+1 ) + qx a(x) ex = ex: n + n