Section 005 date, time & location: Monday 9th, 3.30pm–5.30pm; MSB 315
Section 006 date, time & location: Wednesday 11th, 10.30am–12.30am; MSB 319
Format: 7 questions (you will be expected to attempt all 7). There are no multiple choice
Material covered: The exam will be cumulative, but see below for more details.
Breakdown: 4 of the 7 questions on the exam will cover post-Exam 2 material. The other
three will come from earlier material.
The best way to study is to solve as many problems as possible. In particular
1. You should first make sure that you have completed and understood all of the assigned
2. You should also make sure that you have worked through all of the worksheets (solu-
3. You should then attempt as many other problems from the textbook as you can. As a
general rule, exam questions tend to be similar to homework problems and are often(although not always) taken from the textbook.
4. You should know your basic definitions and results. For more on this see below.
The list below is not intended to be comprehensive, but if you have mastered the topicsbelow, then you should be well placed to tackle the problems in the exam. It is a good ideato take a look at the ‘Summary’ sections at the end of the chapters we have covered.
• Counting arguments are essential to solving many of the problems we have en-
countered in the course. As such, a certain proficiency in combinatorial analysisis essential. You should, by now, have much practice in this. You should, at aminimum, be familiar with
(b) factorials, binomial coefficients, multinomial coefficients and their uses
(a) sample spaces and events, the axioms of probability and some of their basic con-
sequences (e.g. the inclusion-exclusion identity)
(b) computing probabilities in uniform sample spaces (i.e. in a sample space where
all outcomes are equally likely, then P (E) = #E )
(c) the notion of the limit of an increasing/decreasing sequence of events and the fact
that limn→∞ P (En) = P (limn→∞ En)
(d) conditional probabilities, the multiplication rule and Bayes’ formula (and how to
(a) mass functions / density functions and cumulative distribution functions
(b) how to compute the expected value and the variance of a random variable given
(c) how to compute the expected value of a function of a random variable
(d) how to compute the distribution function given the density / mass function and
(e) how to compute the density function of a function of a random variable
(f) how to compute the expectation of a sum of random variables
(g) You should, in general, be able to answer basic probabilistic questions (i.e. P (a ≤
X ≤ b)) using the mass / density / distribution function
(h) you should know the basic information (mass / density function, distribution
Continuous Uniform, normal and exponential
and you should know how to use these to solve problems. You are not expectedto recognise any other types of random variable by name.
(i) You should know how to approximate binomial distributions by a Poisson distri-
bution and by a normal distribution (the latter is, of course, a special case of thecentral limit theorem)
(j) You should know how to work with a Poisson processes and know about the
relationship between waiting times between events of a Poisson process and ex-ponential random variables
(a) joint mass / density functions, joint distribution functions
(b) how to compute the marginal density functions given the joint density function
(c) how to compute the density function of a function of two random variables X and
(d) independence for random variables & how to check it
(e) expectation, expection of functions of X and Y , expectation of products of func-
tions of X and Y when X and Y are independent
(f) covariance (and how to compute it), variance of the sum of independent X and
(a) moment generating functions (you should at least know the definition, be able to
compute the moment generating function in simple cases, know about momentgenerating functions for sums of independent variables and be able to use momentgenerating functions to actually find the moments of a random variable)
(b) limit laws: you should know the statement of the Central Limit Theorem and
Since we covered the later material rather quickly (and rather recently!) let me give yousome more specific information on this. You can expect:
1. A question which requires you to use the Central Limit Theorem to approximate a
2. A question asking you to compute Cov(X, Y )
3. A question asking about the independence of two random variables X and Y
4. A question involving using moment-generating functions to compute moments
I expect your grasp of the earlier material (Chapters 1–5) to be more comprehensive, so
I will offer no further guidance there.
Good luck in all of your exams this year!

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