Is being the first-born independent of age? If one were to assert that, in a large population, the fact of a person or an animal being the first-born in the family was independent of their age, then what assumptions (if any) would one have to make about the population?
This question arose when using the ten plagues of Egypt for an example of survival analysis.
 A: I propose a qualitative answer.
The idea is to equate the first-born rate at age A with the ratio between the size of the population of first-born (younger) parents and that of second-born (older) parents at time T=(now - A years).
If you make the assumption that the number of children per adults and the age when they get their children does not change over time, I would say that the age and  being first-born are independent.
To prove it, you can consider the number of births at time T of first-born children per day at any given date. It is proportional to the number of their parents : size(P1(T)) while the number of births of other children is proportional to the number of their parents : size(P2(t)).
It is obvious that the population P1(T) is younger than P2(T)
So, as the population is, by hypothesis, increasing at a geometric pace, size(P1(T))/size(P2(T)) is not changing with time. Therefore, so it is for the rate of first-birth over total birth. Considering that the probability to be first-born child when your age is A is the rate of first-birth A years ago , you can conclude that it does not depend on A.
It is interesting to remark that this rate is not necessary the inverse of the average number of children per couple. In case of population growth, there is a bias toward first-born children.
Conversely, if the fertility rate (or the age when people get children) change over time, there would be change in the ratio size(P1(T))/size(P2(T)) along T. So the 2 variables considered will become dependent.
