The mathematics Life table
tpx chart table 1. life table total population: united states, 2003, page 8
the basic algebra used in life tables follows.
q
x
{\displaystyle \,q_{x}}
: probability aged
x
{\displaystyle \,x}
die before reaching age
(
x
+
1
)
{\displaystyle \,(x+1)}
.
p
x
{\displaystyle \,p_{x}}
: probability aged
x
{\displaystyle \,x}
survive age
(
x
+
1
)
{\displaystyle \,(x+1)}
.
p
x
=
1
−
q
x
{\displaystyle \,p_{x}=1-q_{x}}
ℓ
x
{\displaystyle \,\ell _{x}}
: number of people survive age
x
{\displaystyle \,x}
note based on radix., or starting point, of
ℓ
0
{\displaystyle \,\ell _{0}}
lives, typically taken 100,000
ℓ
x
+
1
=
ℓ
x
⋅
(
1
−
q
x
)
=
ℓ
x
⋅
p
x
{\displaystyle \,\ell _{x+1}=\ell _{x}\cdot (1-q_{x})=\ell _{x}\cdot p_{x}}
ℓ
x
+
1
ℓ
x
=
p
x
{\displaystyle \,{\ell _{x+1} \over \ell _{x}}=p_{x}}
d
x
{\displaystyle \,d_{x}}
: number of people die aged
x
{\displaystyle \,x}
last birthday
d
x
=
ℓ
x
−
ℓ
x
+
1
=
ℓ
x
⋅
(
1
−
p
x
)
=
ℓ
x
⋅
q
x
{\displaystyle \,d_{x}=\ell _{x}-\ell _{x+1}=\ell _{x}\cdot (1-p_{x})=\ell _{x}\cdot q_{x}}
t
p
x
{\displaystyle \,{}_{t}p_{x}}
: probability aged
x
{\displaystyle \,x}
survive
t
{\displaystyle \,t}
more years, i.e. live @ least age
x
+
t
{\displaystyle \,x+t}
years
t
p
x
=
ℓ
x
+
t
ℓ
x
{\displaystyle \,{}_{t}p_{x}={\ell _{x+t} \over \ell _{x}}}
t
∣
k
q
x
{\displaystyle \,{}_{t\mid k}q_{x}}
: probability aged
x
{\displaystyle \,x}
survive
t
{\displaystyle \,t}
more years, die within following
k
{\displaystyle \,k}
years
t
∣
k
q
x
=
t
p
x
⋅
k
q
x
+
t
=
ℓ
x
+
t
−
ℓ
x
+
t
+
k
ℓ
x
{\displaystyle \,{}_{t\mid k}q_{x}={}_{t}p_{x}\cdot {}_{k}q_{x+t}={\ell _{x+t}-\ell _{x+t+k} \over \ell _{x}}}
μx : force of mortality, i.e. instantaneous mortality rate @ age x, i.e. number of people dying in short interval starting @ age x, divided ℓx , divided length of interval.
another common variable is
m
x
{\displaystyle \,m_{x}}
this symbol refers central rate of mortality. approximately equal average force of mortality, averaged on year of age.
further descriptions: variable dx stands number of deaths occur within 2 consecutive age numbers. example of number of deaths in cohort recorded between age of 7 , age of eight.
the variable ℓx, stands opposite of dx, represents number of people lived between 2 consecutive age numbers. ℓ of 0 equal 100,000. variable tx stands years lived beyond each age number x members in generation. Ėx represents life expectancy members @ specific age number.
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