Can you show us the model that you have chosen to use for this problem? In general, your model could be as complicated as you wanted with all kinds of joint distributions taking any pieces of information it wants to.
You have written the model g (Y(t-1)) =SUM(Xi from i=1 to Y(t-1)). Is this the one you want to use?
If so, you should probably look at a compound Poisson process as well as probability generating functions for Bernoulli processes. You might also want to have a look at generalized linear models and link functions for your g().
I'd check out compound poisson processes first though. Have you covered them in your course?