Rate Interval

If the Death data has the rate interval of age nearest birthday
why does the estimate of the central rate of mortality give us the value for x-1/2 to x+1/2?

Shouldn't we treat it as constant force of mortality and hence shouldn't it estimate the value at x only?

 

Sorry, what I mean to say is that shouldn't we consider the rates to be uniform and hence it would estimate the value at x ?

May 2008 of IAI has a question regarding the same. Q3
It says that mu should estimate the value from x-1/2 to x+1/2

 

Ok, thank you very much.

There was another question on rate intervals from the same paper which i found to be difficult:
IAI MAY 2008 Q5
A life insurance company issues only annual premium life assurance policies. The company keeps records of its life assurance policies in two files; inforce file and a claims file. For each premium paying policy, the in-force file includes the following information:

• age last birthday at the date of policy issuance
• smoking status (smoker or non-smoker)
• sex
• type of policy (temporary assurance, whole life assurance or endowment
assurance)
For each age x (where x = age last birthday at date of policy issue + number of annual premiums paid since policy issuance) a count of the number of policies sub-divided by smoking status, sex and type of policy is tabulated on 1st January of each year.
For each policy for which a death claim has been paid the claim file contains the same information as the inforce file.
For each age x (where x = age last birthday at date of policy issuance + number of annual premiums paid up to the date of death) a count of death claims divided by smoking status, sex and type of policy is tabulated on each 1st January.
The company wishes to investigate the recent mortality experience of its life assurance policies
Defining suitable symbols, derive a formula for the estimation of the force of mortality using the data for an age group x according to the definitions given. State any assumptions that are required for your formula. State, with reasons, the age to which your estimate of the force of mortality applies.