In tutorial 4 question 2 ii (attached below) we were told to estimate the force of morality and initial rate of mortality using the UDD assumption. So I calculate \(\hat{u_x}\) which is an estimate for \(u_{x+0.5}\). Now to get an estimate for \(q_x\) I can either use the relationship which we prove in question 1 OR I assume that \(u_{x+t}\) is linear between ages x and x+1 and solve using $$ q_x = 1 - exp(-{\int {u_{x+t}dt}}) = 1 - exp(-0.5*(u_{x}+u_{x+1})) = 1 - exp(-u_{x+0.5}) = 1 - exp(- \hat{u_x} )$$

which is what I did in the tutorial test and was marked correctly, but the relationship proven in question 1 contradicts the way I did it. If you could shed some light on what I am missing I would really appreciate it.

Thanks for the answer, it makes a lot more sense now. However, I'm confused as to why we even make the linearity of \(u_{x+t}\) assumption if we already have this nice relationship that is more precise and easier to use. Like I get that it is an estimate, but why further muddle up the estimate with more assumptions when the UDD assumption is more than enough?

The estimate for \(u_{x}\) will be slightly different when making the linearity assumption and the UDD assumption compared to just making the UDD assumption, but we can get this better more precise estimate with less work and less assumptions, so I would think that we should just make the UDD assumption, right?

Thanks