Thank you for the question. I do not seem to agree with the age to which you say the force of mortality applies to.

You are correct in saying that the death data specifies the rate interval. **Now, there are TWO STEPS to find the age to which the force of mortality applies to**:

1. Calculate the age at the beginning of the rate interval. This we infer from the AGE LABEL of the deaths, which is "age x nearest birthday at the point of exposure" (the time we want to age the life, in this case it is death). So, the life maintains this age label from 6 months before their xth birthday, to 6 months after. Hence, the EXACT age at the start of this rate interval is \(x - \frac{1}{2}\). Please will you now write out the rate interval clearly.

2. Then, find the age to which \(\mu\) applies to - recall that we said that (in most cases) \(\mu\) applies to the average age at death in the rate interval, and if we **assume** that deaths occur on average half way through the rate interval, then that age is \(x\).

Fortuitously, we obtain the estimate for \(x\).

Now, to adjust the census data. The first thing to note about this question is that you are given EXPOSURE data (ie. times of exposure) and NOT POPULATION DATA. I will give you a hint to get you started.

We have \(E_x^C\ast\) , which is the years of exposure during the period of investigation, for lives aged \(x\) next birthday. We wish to extract information on \(E_x^C\) , which is defined to be the years of exposure during the period of investigation, for lives aged \(x\) nearest birthday. Now, of the lives contributing exposure to \(E_x^C\) , 1/2 will be between the ages (x-1/2,x), and the others between ages (x, x+1/2) assuming that birthdays are uniformly distributed over the year which starts 6 months before the time point of exposure (i.e. your date of census, which is NECESSARY since we have that \(E_x^C\) is a function of \(P_x(t)\) and \(P_x(t+1)\) <- here, your census dates are t and t+1), to 6 months after (a subtle concept - draw a timeline to see this!). On the basis of this information, I hope you can now extract the relevant exposure for the lives age $x $next.

Hi Mario,

Thank your very much for your comprehensive reply, particularly on a weekend.

I seem to have gotten myself confused with regard to the way in which we tackled to which age the force of mortality applies for the rate intervals defined by

"age nearest birthday at previous policy anniversary (PA)",where we assumed that qx applies to average age x and with the force of mortality applying to age x+1/2. Is it correct to say that the nuance of having to assume a UDB (as for the PA RI) falls away in our case in this question, since for the life rate interval here we know that the life will hold the age label from exactly x-1/2 to x+1.2 as you mentioned? So the life rate interval is based on the individual's ageing based on his birthday rather than the PA.// Hope that was decipherable, quite tricky to articulate some of these ideas out loud //