Ch14 Page 8 of Notes

[Snowy]

The probability of a life aged x-1 last birthday at n-1 dies in (n-1,n), assuming uniform distribution of births over the calendar year is qx-1/2 (n-1).

Can we assume uniform distribution of deaths here and the probability will be 1/2qx-1(n-1)?

Why does the notes use the uniform distribution of births assumption rather than the usual death assumption?

 

[Robert Chadburn]

hi Snowy
Those aged x-1 last birthday at time n-1 are aged between x-1 and x at that time. So we are saying that the average age of these people is x-1/2 at time n-1, and the only way we can say this is if we assume their birthdays are uniformly distributed with respect to calendar time, ie their x-1th birthdays are evenly spread over the calendar year. The probability now boils down to qx-1/2(n-1), which is the probability that a person aged x-1/2 at time n-1, dies over the next year.

On the other hand, the probability you quote is 1/2qx-1(n-1). Well, qx-1(n-1) is the probability that someone aged x-1 dies over the year, so if you take half of this then you (approximately) have the probability that someone aged x-1 dies over HALF a year, and that is NOT what is required.

The way that deaths are distributed is irrelevant to the problem, as all we are needing is an approximation to the starting age of people at the beginning of the year.

I hope this helps.