Comprehensive Exams, Statistics

Ph.D. Preliminary Examinations

The Applied Maths Version and Syllabi are omitted from this document.

The Pure Maths Version and Syllabi are omitted from this document.

The Department of Mathematics and Statistics requires all doctoral students to pass two preliminary examinations, first the Part A and then the Part B. The Part A examination has to be passed by the end of the first year of registration at the PhD 2 level. All graduate students thinking in terms of doctoral studies at McGill are urged to familiarise themselves with the regulations governing these crucial examinations and to plan their studies taking these examinations into account.

MATH 700 — Ph.D. Preliminary Examination Part A.

General Regulations


The Ph.D. Part A examination is a written examination consisting of two papers, which, with the exceptions noted below, must be taken and passed together within the same examination period. There are three versions: Applied Mathematics, Pure Mathematics, and Statistics. In the Applied and Pure versions the papers are called alpha and beta with Paper alpha the same for both versions; in the Statistics version they are called theory and methodology. Each graduate student will be examined in one of these versions and the choice is made by the student in consultation with the Advisory Committee of the student. Two thirds of Paper alpha covering topics in real analysis and linear algebra at the level of our first year undergraduate honours courses is traditionally referred to as the compulsory part.

Deadline to Pass the Part A Examination

This examination has strict deadlines. Only students who have successfully passed the Part A examination will be permitted to register for the September semester of the Ph.D. 3 year.

Schedule of Examination

Each graduate student is allowed only two official attempts at the Part A examination. The examination will be administered twice a year, once in May and once in August. The parts alpha and beta, or theory and methodology, are held on separate days within one week. So under usual circumstances, a student will make the first attempt in May preceding September of the Ph.D. 3 year, so that if necessary, a second attempt can be made before the September deadline, that is, in August. However students may take the examination earlier if they feel ready for it and scheduling permits. Notices of the precise time and place of the examination are posted a few weeks beforehand. A student intending to write the Part A examination must notify the Department office in writing of theit intention by March 30 for the May examination and June 30 for the August examination. Each paper will involve a 4-hour sitting. The papers will be set in such a way that a strong candidate can be expected to complete the paper within 3 hours.


Normally, a student who fails at the first attempt will be required to re-take and pass the entire examination, although in exceptional cases the committee may allow the student to take only one part . However, this permission must be sought and given before the examination. Also, in case of exceptional circumstances such as illness, the Committee of Graduate Affairs (CGA) may relax the deadline to pass the Part A examination as seems appropriate.

Free Trials

Students are allowed one free attempt at the examination. "Free" here means that in case of failure, the student still has the 2 official attempts (see Schedule of Examination), whereas success is just as valid here as in the official attempts. This attempt, however, must be made before registration in the September semester of the Ph.D. 2 year. This gives the opportunity to Master's and Ph.D. 1 students to take the examination on a variety of occasions, but also includes the possibility that a Ph.D. student admitted at Ph.D. 2 level (which is the usual case) to take a free trial in August, before registering for the September semester of this Ph.D. 2 year. Note that a student enrolled in a Master's program, who passes a free trial examination is entitled to apply for direct promotion to Ph.D. standing without getting a M.Sc. degree. In any case, a student who passes such a free try has fulfilled the Part A requirement. For these reasons we cannot stress enough the benefits for our Master's students of making a serious free attempt at the Part A examination. in anticipation of later enrolment at the Ph.D. level.


The Part A Committee will recommend to the CGA the results of the examination. The result will be reported as PASS/FAIL. While the evaluation will be based on the overall performance of the student on both papers, candidates taking the Applied or Pure examinations will be expected to perform at least satisfactorily on the compulsory part of Paper alpha. There will be no partial passes but the examiners may recommend a specific additional requirement (such as grading or taking certain courses) to be fulfilled by individual students who have displayed a conspicuous weakness. The CGA will present the results of the examination to the Department and the results may be appealed to the Department by the student or by any member of the Department. However, the student will not be permitted to register, pending an appeal of a failure. A student who writes the Part A exams in May and fails the compulsory (undergraduate) part of paper alpha, but has an overall passing performance, may, at the recommendation of the Part A Subcommittee, be given the option to sit one 4-hour exam on the compulsory material of paper alpha. Regarding deadlines, this option would be treated the same way as a full attempt at the Part A exams.

Statistics Version

The Statistics Part A exam consists of a Theory Paper (in mathematical statistics and measure theoretic probability) and a Methodology Paper (in linear models and generalized linear models).

The exam format has been thought out so that a student who has training in linear regression and has taken Mathematical Statistics 1 (MATH 556), Advanced Probability Theory 1 (MATH 587), Mathematical Statistics 2 (MATH 557), Generalized Linear Models (MATH 523) during their first year has all the requisite notions fresh on her or his mind by May. The best time to attempt them is thus at the end of the first year.

Syllabus for the Theory Paper — Statistics

Measure, Integration and Foundations of Probability: Open, closed and compact sets in Rk. Basic properties of probability spaces: s algebras, axioms of probability functions and calculus of probability. Outer measure, measure extension theory and Lebesgue measures on Rk. Measurable mappings, limits and measurability of functions. Distribution functions and expectations: Chebychev's inequality, Markov's inequality and Jensen's inequality. Moment generating functions; Stieltjes integrals on Rk; Uniform integrability; Lebesgue's dominated convergence theorem, monotone convergence theorem and Fatou's lemma. Independence. Product spaces, product measure and Fubini theorem. Integration by parts. Almost sure convergence, convergence in probability and weak convergence. Law of large numbers; Kolmogorov's zero-one law. Basic properties of the Poisson process. Definitions and basic properties of conditional expectation and conditional probability.

Advanced Probability Theory 1 (MATH 587)
Reference: Billingsley, P. (1986). Probability and Measure. 2nd ed. Wiley. [Unstarred sections of Chapters 1 to 4.]

Mathematical statistics: The calculus of probabilities. Counting. Equally likely outcomes. Conditional probability and independent random variables. Distribution Functions: Density and mass functions. Expected values. Moments and moment generating functions. Common families of distributions: exponential families, location and scale families. Joint and marginal distributions. Conditional distributions and independence. Multivariate transformations. Covariance and correlation. Mixture distributions. Order statistics. Sums of random variables from a random sample. The Lindeberg-Levy Central Limit Theorem. The sampling distributions: Student's t and Snedecor's F. Sufficiency, minimal sufficiency, ancillarity, completeness, Basu's Theorem, Rao-Blackwell Theorem, Lehmann-Scheff Theorem, minimum variance unbiased estimation. Method of moments, maximum likelihood estimation, Bayes' estimation, invariant estimation, consistency. Asymptotic properties of maximum likelihood estimators, the "delta method" for functions of random variables. Hypothesis testing: concepts of significance and power, the Neyman-Pearson lemma, likelihood ratio tests, Bayesian tests. Interval estimation: methods of finding confidence intervals including inverting a test statistic and pivotal quantities. Bayesian credibility intervals.

Courses: Mathematical Statistics 1 (MATH 556), Mathematical Statistics 2 (MATH 557)
Reference: Casella, G. & Berger, R. L. (2001). Statistical Inference. 2nd ed. Wadsworth [Chapters 1-10].

Syllabus for the Methodology Paper — Statistics

Linear models: Least squares estimators and their properties. Simple linear regression, multiple regression, Gauss-Markov theorem. Analysis of variance. Linear models with general covariance. Distribution of estimators. General linear hypothesis: F-test and t-test, prediction and confidence regions. Categorical and continuous covariates, collinearity, interactions. Multivariate normal and chi-squared distributions. Model selection, residual analysis, detecting influential observations, testing for lack of fit, transformations, weighted least squares, and variable selection techniques.

Course: Regression and Analysis of Variance (MATH 423)
Weisberg, S. (1980). Applied Linear Regression. New York: Wiley.

Generalized linear models: Exponential families. Link function, variance function. Iteratively reweighted least squares, asymptotic distribution of maximum likelihood estimators. Analysis of deviance, goodness of fit tests. Log-linear models, analysis of contingency tables, overdispersion. Logistic regression, case-control studies, multinomial regression. Gamma models. Quasi-likelihood models. Mixed models. Bayesian estimation.

Course: Generalized Linear Models (MATH 523)
: McCullagh, P. & Nelder, J. (1989). Generalized Linear Models. 2nd ed. Chapman & Hall: New York.

Old Examinations

Old examinations are available in the library; they go back until 1984. Some of the more recent examinations are available on-line. Although the syllabus and format of the examinations have changed, they still give an idea of what to expect. The more recent examinations are closer to the present syllabus. Note that starting in May 2007 there have been major changes in the applied mathematics version of the exam and minor changes in the pure mathematics version.

MATH 701 — Ph.D Preliminary Examination Part B.

  1. The purpose of the Part B Examination is to ensure that a student getting a Ph.D. degree from our Department be well versed in at least two areas of Mathematics and Statistics which go beyond the basic knowledge required for the Part A Examination.
  2. A Standing Committee on the Part B Examination shall be set up by the Graduate Affairs Committee and shall consist of a Chair and two other members. This subcommittee will be responsible for approving the choice of examination topics for each candidate, for designating the members of the student's Examining Committee and for arranging the place, date and time of the examination. The Part B Subcommittee will assume total responsibility for informing students of deadlines both impending and expired concerning the Part B Examination.
  3. The student, in consultation with the supervisor, will propose to the Part B Subcommittee two topics and a principal examiner for each. Each topic should be based on a one semester 600 or 700 level course, or similarly advanced material, and will usually be defined by a book, a designated part of a book or a collection of research papers. A reasonably detailed description, countersigned by the prospective principal examiner, will be submitted in writing to the Part B Subcommittee. The Part B Subcommittee will inform the student in writing of its approval of the proposals. The exam is to be held not earlier than 2 and no later than 6 months from this date. The student will inform the Part B Subcommittee at least one month in advance of the date the exam is to be taken. The Part B Subcommittee will then set up an examining committee consisting of the 2 principal examiners, two other examiners, and a member of the Part B Subcommittee who will act as Chair of the examining committee.

    The exam will be open to members of the Department. The exam will be conducted mainly by the principal examiners. The Part B examination is an oral examination: notes, books, or other materials are not allowed. After the principal examiners have finished their part, the other members of the examining committee and other members of the Department present may pose further questions. The student's status at the end of the exam is to be determined by a majority vote of the examining committee.

  4. The student must take a Part B Examination not later than five semesters (not including summers) after beginning the Ph.D. program. If the student fails in one or both topics presented, he or she may repeat the examination once, and this within the following four months, at a time to be approved by the Part B Subcommittee.
  5. The Part B Examination must be completed successfully before scheduling the thesis defense.
  6. These regulations are to take effect beginning in September 1997.
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