This course provides a structured development of probability theory, especially the theory of random variables and random vectors. The pace of the course is brisk, as it begins from the assumption that students have little prior exposure to probability yet reaches advanced concepts by the end.

20 lectures

5 tutorial hours

STATS2002 Statistics 2R: Probability

STATS2005 Statistics 2X: Probability II

90-minute, end-of-course examination (80%)

coursework (20%)

The aims of this course are: 

■ to provide a structured development of probability theory, with an emphasis on the theory of random variables and random vectors;

■ to prepare students to solve real-life problems using probability theory.

By the end of this course students will be able to:

■ state the Axioms of Probability and use them to prove basic propositions in probability theory;

■ use probability mass functions and probability density functions in one or more dimensions to compute probabilities and percentiles in particular cases;

■ write down general definitions of moments in one or more dimensions (including the vector of expected values and the variance-covariance matrix), derive general properties from these definitions, and compute moments in particular cases;

■ recognise standard discrete and continuous probability distributions in a context, and use them to obtain probabilities, percentiles and moments;

■ use the joint distribution of a random vector to derive marginal or conditional distributions of one or more of the component variables;

■ determine whether two or more random vectors are independent;

■ write down general definitions of the probability generating function, moment generating function and characteristic function in one or more dimensions, derive general properties from these definitions, and compute the functions in particular cases;

■ find the distribution of functions of random variables in one or more dimensions;

■ use standard methods to derive the exact distribution of the sum of a sequence of random variables;

■ state and use the laws of large numbers and the central limit theorem;

■ state and use properties of the multinomial and Multivariate Normal (MVN) distribution;

■ integrate their knowledge of topics in the course to solve realistic problems. 

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.