Consider a population containing smokers ((s)) and non-smokers ((ns)) where their forces of mortality are related as
begin{eqnarray*}
mu_x^{ns} = frac{1}{2} mu_x^s .
end{eqnarray*} For the non-smoking population, mortality is given as (l_x = 75 – x) (xge 0). Consider two lives, a non-smoker age 65 and a smoker age 55, and assume lives are independent. Calculate (dot{e}_{65:55}).
frac{-d~l_x}{dx} frac{1}{l_x} = frac{1}{75-x}). Thus, the
conditional survival force is (~_t p_x^{ns} = 1 – frac{t}{75-x}).
For smokers, the force of mortality is (mu_x^s = frac{2}{75-x}) so that the conditional survival force is
begin{eqnarray*}
~_t p_x^s &=& expleft( – int_0^t mu_{x+r}^s ~ dr right) \
&=& exp left( -2 int_0^t frac{1}{75-x-r} ~ dr right) \
&=& exp left( 2 ln left(75-x-r right)|_0^t right) \
&=& exp left( 2 ln frac{75-x-t}{75-x} right) \
&=& left(1- frac{t}{75-x} right)^2
end{eqnarray*}
Thus,
begin{eqnarray*}
~_t p_{65:55} &=& ~_t p_{65}^{ns} ~_t p_{55}^s = left(1-
frac{t}{10} right)left(1- frac{t}{20} right)^2
end{eqnarray*} and
begin{eqnarray*}
dot{e}_{65:55}&=& int_0^{10} ~_t p_{65:55} dt \
&=& int_0^{10} left(1- frac{t}{10} right)left(1- frac{t}{20}
right)^2 dt = 3.54 .
end{eqnarray*}
As follow-ups, note the following extensions/observations.
1) (dot{e}_{65}^{ns} =5 (=frac{75-65}{2} )).
2) (dot{e}_{55}^s = int_0^{20} left(1- frac{t}{20} right)^2 dt
= frac{20}{3}).
3)
begin{eqnarray*}
dot{e}_{overline{65:55}} &=& 5 + frac{20}{3} – 3.54 = 8.12\
& neq & int_0^{10} ~ _t p_{overline{65:55}} ~dt .
end{eqnarray*}