Probability, Distribution Theory and Inference

This information is for the 2022/23 session.

Teacher responsible

Dr Miltiadis Mavrakakis-Vassilakis


This course is compulsory on the BSc in Actuarial Science and BSc in Financial Mathematics and Statistics. This course is available on the BSc in Business Mathematics and Statistics, BSc in Data Science, BSc in Econometrics and Mathematical Economics, BSc in Mathematics and Economics, BSc in Mathematics with Economics and BSc in Mathematics, Statistics and Business. This course is available as an outside option to students on other programmes where regulations permit and to General Course students.


Students must have completed Elementary Statistical Theory (ST102) and Mathematical Methods (MA100).

Students who have not taken these courses should contact Dr Mavrakakis.

Students must have completed one of the following two combinations of courses: (a) ST102 and MA100, or (b) MA107 and ST109 and EC1C1. Equivalent combinations may be accepted at the lecturer’s discretion.”

Course content

The course covers the probability, distribution theory and statistical inference needed for advanced courses in statistics and econometrics.

Michaelmas term: Probability. Conditional probability and independence. Random variables and their distributions. Moments and generating functions. Transformations. Sequences of random variables and convergence. Multivariate distributions. Joint and marginal distributions. Expectation and joint moments. Independence. Multivariate transformations. Sums of random variables. Conditional distributions. Conditional moments. Hierarchies and mixtures. Random sums.

Lent term: Random samples. Sample mean.  Sampling from the Normal distribution. Order statistics. Sample statistics. Sampling distributions. Parameter estimation. Interval estimation. Hypothesis testing. Maximum-likelihood estimation.  Likelihood-ratio test. Sufficiency and minimal sufficiency. Rao-Blackwell theorem. Cramér-Rao lower bound. Most powerful tests. Neyman-Pearson lemma. Linear regression. Least-squares estimation. Generalised linear models.


This course will be delivered through a combination of classes and lectures totalling a minimum of 60 hours across Michaelmas Term and Lent Term. In addition to these, there will be (optional) weekly workshops to help with homework assignments. This course includes reading weeks in Week 6 of Michaelmas Term and Lent Term.

Formative coursework

Students will be expected to produce 4 pieces of coursework in the MT and LT.

These are exam-style class tests.

Indicative reading

M C Mavrakakis & J Penzer, Probability and Statistical Inference: From Basic Principles to Advanced Models (primary reading)

G C Casella & R L Berger, Statistical Inference (very useful as a reference)


Exam (50%, duration: 2 hours) in the January exam period.
Exam (50%, duration: 2 hours) in the summer exam period.

Student performance results

(2019/20 - 2021/22 combined)

Classification % of students
First 43.7
2:1 30.1
2:2 17
Third 6.6
Fail 2.6

Key facts

Department: Statistics

Total students 2021/22: 180

Average class size 2021/22: 28

Capped 2021/22: No

Lecture capture used 2021/22: Yes (MT & LT)

Value: One Unit

Guidelines for interpreting course guide information

Course selection videos

Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.

Personal development skills

  • Problem solving
  • Application of information skills
  • Application of numeracy skills
  • Specialist skills