## MIDTERM

The midterm is a  closed book exam which will be held in class. It will cover class notes as indicated on the website and assignments 5,6,7.

Here is a practice exam from 2012.   Ignore questions about N(t) and the Poisson process because we have not covered these  topics.

Don't try to read all the examples in the book; use your class notes as a guide on which ones to understand and compare the way the book explains theorems and definitions with how I did it in class and try to reconcile the two.

There will be four questions. Each may have several parts and some parts may be checking if you know definitions and theorems.  For example

Let (X_n, n >= 0) be a branching process.  Define the generating function G_n(s) for X_n.   [G_n(s) = E s^{X_n}]

What is G_n(s) at s =0?  [Prob X_n = 0]

What formula relates G_{n+1}(s) to G_n(s)?    [G_{n+1}(s) = G( G_n(s) ) where G(s) = Es^{Z} where Z  is the number of children of an individual]

State the no-memory property of an exponential random variable
[ P(T>t+s|T>s) = P(T>t) ]

Prove the no-memory property   [done in class, uses P(T>t) = e^{-lambda t}]

You do not have to learn long proofs, but you should understand important ideas in proofs such how the moment generating functionis related to  moments and why the generating function of a sum of independent random variables is easy to calculate.

There will also be questions with a problem to solve. These will be just like homework problems; they may even be homework problems but with changes in numbers, so make sure you understand the solutions to the homework.

• There will be no need for a calculator.

• Bring photo ID and put it out in case I want to check it.

• Arrive five minutes before the hour,  put coats, backpacks, books ... at the front of the room. Do not wear earphones for ipods etc.

• Sit so that there is as much distance as possible between you and your neighbours.

• When you are asked to stop writing at the end of the exam, we truly expect you to STOP WRITING so that we can get out of the room on time. If you continue, we might give you a reduced mark or even not mark your exam.