Module 1 Flashcards
S(x) =
= 1 - F(x)
= Pr( X>x)
Tx is the
Future lifetime, of an individual already aged x
Fx(t) =
=Pr[Tx ≤t]
=Pr[T≤ x+t | T>x]
=[S(x)−S(x+t)] / S(x)
other ways of writing tqx
= Pr(Tx < t)
= Fx(t)
= from 0 to t
∫ fx(s) ds
other ways of writing tpx
= Pr[Tx > t]
= 1 - tqx
= Sx(t)
Deferred probability meaning
n|mqx
Age x, death after the next n years but within n+m years
Pr[ nx
s+tpx
can be re-written as
(tpx)(spx+t)
tpx in terms of survival function
Sx(t)
µx force of mortality is defined as
= limh->0 (1/h) * Pr[T ≤ x+h | T>x]
= f(x) / S(x)
=S’(x) / S(x)
Cumulative hazard function
H(x) =
from 0 to x
∫ µt dt
= -log[S(x)]
S(x) in terms of hazard function
= exp( -∫µt dt )
from 0 to x
tpx
in terms of hazard function
= exp( -∫µs ds )
from x to x+t
Important result for
fx(t) =
= tpx * µx+t
Kind of like the chance of making it to t then instantaneously dying.
ėx is the
Complete expectation of life aged x
ie E[Tx]
ėx
in integral form is
∫ tpx dt
from 0 to w-x
recall tpx = Sx(t)
What is the curtate future lifestyle (Kx) of age x
The integer part of Tx
Pr[Kx =k]=
in p and q forms
=Pr[ k≤ Tx
=kpx • qx+k
curtate expectation of life ex is defined by
=E[Kx]
= A big complicated sum thing, see note.
npx
in terms of lx
lx+n / lx
dx in terms of lx
lx - lx+n
nqx
in terms of d and l
ndx / lx
µx in terms of S(x) or lx
= -S’(x) / S(x)
= - lx’ / lx
n|mqx
in terms of lx
( lx+n − lx+n+m ) / lx
What is initial rate of mortality
Prob a life aged x dies before age x+1
denoted qx
