I've been stuck on this question for the past half hour and still have no idea how to solve it... I don't think it's supposed to be very difficult but I'm struggling:
There are two independent live which are identical except that one is a smoker and the other is a non-smoker. It is know that:
$\mu_x$ is the force of mortality for non-smokers for $0\le x<\omega$
$c\mu_x$ is the force of mortality for smokers for $0\le x<\omega$, where c is a constant, $c>1$
Calculate the probability that the remaining lifetime of the smoker exceeds that of the non-smoker and check for the reasonableness of your answer.
Additional details:
To solve the problem we're supposed to denote Z as the age at death with cdf $F_Z(x)$ and pdf $f_Z(z)$ and the survival function defined as $s(x)=1-F_Z(x)$
The force of mortality represents the likelihood of the individual aged x dying in the next instant dt i.e. $$\mu_x=\lim_{\Delta x \to 0}\frac{F_Z(x+\Delta x)-F_Z (x)}{\Delta x(1-F_X (x)) }=\frac{f_Z(x)}{1-F_Z (x) }=\frac{-s' (x)}{s(x)} $$
c is a constant and $\omega$ denotes the maximum possible age attained so $F_Z(\omega)=1$
We also use $T(x)=Pr(Z-x | x>0)$ to denote the future lifetime
I have not really made much meaningful progress however.
Edit: Sorry I almost forgot I think the force of mortality should be uniformly distributed between $x$ and $\omega$... i.e. $\mu_{x+t}=\frac{1}{\omega-x-t}$