So to answer your question, I first need to emphasise one of the most important things in exposure to risk. **RATE INTERVALS**. While the question does not explicitly ask the question, you should always be thinking of this and define them for your own use. Without defining them, you've basically put yourself in an impossible situation to answer the question. While I do know the rate intervals for deaths and the census (from my own definitions), I would prefer you to do it yourself and then rather compare your attempted answer, then simply reading my answer.

So to answer your questions:

1. *"does that mean that dx will hold the number of deaths aged x last birthday at death for the 12 months prior to the census, with half of these being deaths that occurred before the life could celebrate a birthday this year and the other half being deaths where the life turned x this year and then died?"*

No, the definition of dx does not imply this (even after making those assumptions). You've incorrectly come to this conclusion due to an incorrect definition (implicit) of the rate interval for deaths. This make's the second half of the question (*"Extending this further if it is true, does this mean that at the time of census, half of the deaths in dx would be aged x+1 last birthday had they not died?"*) irrelevant, as it is untrue.

2. So you seem to be thoroughly confused regarding the definition of dx in the question. **dx represents the number of deaths of lives classified as age x (according to the defined rate interval - try figure this out yourself first) that have died in the 12 months prior to the census date.** As a result, your question *"*if we wanted the number of deaths in the year preceeding the census aged x last birthday at the time of census" implies a different rate interval, and it would indeed be a different dx to the one defined in the question. This brings me to another critical point of Exposure to Risk, which I mentioned in the tutorial last week, **We do not manipulate death data.** Death data, when given, is usually very concise and difficult to make reasonable assumptions about to calculate deaths over a different rate interval. Thus we usually concern ourselves with manipulating census and exposure data.

I hope that helps you with this question, and allows you to make a reasonable attempt at it. If you still cannot figure it out, comment or ask another question, and I can take you through a rough version of the answer.