 
Summary: Foundations of Stochastic Processes 048868
Final Exam
Due 09:00, July 19, either by hand to Meir 1233 or electronically to
robert@ee.technion.ac.il
The exam might seem long, but, in the unforgettable words of The Hitch
hiker's Guide to the Galaxy,
DON'T PANIC!
The reason each question is so long is that I broke it into small pieces so
that if there is one step that you cannot do you can still go on from there
to the next one.
Question 1: My home is equidistant to a cake shop and a chocolate shop,
and each day I choose, at random, to go to one of these shops to buy either
a cake or a chocolate for breakfast.
On the first day, I have no preference for either shop, and so choose
between them with equal probabilities. However, both the cakes and the
chocolates are very tasty, and so the positive reinforcement I get from my
choice affects my probabilities for later days, in the following fashion:
Let Nn be the number of times that I have visited the cake shop after n
days. Then, given the entire history of the process up to and including day
n, the probability of choosing the cake shop on day n + 1 is
