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
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