Prerequisites

Statistics is a mathematical science that requires practical working knowledge of a number of subfields in mathematics, most importantly in probability theory but also in vector and matrix algebra as well as in the calculus of functions of several variables.

The MATH27720 Statistics 2 module (year 2, semester 2) builds on the earlier mandatory probability and statistics modules offered by the University of Manchester such as Statistics 1 and Probability 1 (year 1) as well as on Probability 2 (year 2, semester 1). Many students will also have attendend the Practical Statistics and the Linear Regression modules (year 2, semester 1).

Below you find a number of resources to help you to refresh your knowledge in these areas.

Matrices and calculus

For a refresher of the essentials in matrix algebra and calculus please refer to the supplementary Matrix and Calculus Refresher notes.

Below you find topics that are particularly relevant.

Vectors and matrices

  • Vector and matrix notation
  • Vector algebra
  • Eigenvectors and eigenvalues, eigendecomposition
  • Properties of special matrices (e.g. a real symmetric matrix has real eigenvalues)
  • Positive and negative definiteness of a matrix (i.e. matrices with only positive or only negative eigenvalues)
  • Matrix inverse

Functions

  • Gradient vector
  • Hessian matrix
  • Conditions for a local extremum of a function
  • Convex and concave functions
  • Linear and quadratic approximation

Logarithms

In these notes the logarithmic function \(\log(x)\) will always refer to the natural logarithm when the base is not explicitly stated. Logarithms with base 2 and based 10 are denoted as \(\log_2(x)\) and \(\log_{10}(x)\), respectively.

Combinatorics, probability and distributions

For a detailed introduction of concepts in combinatorics and probability see the lecture notes for Probability 1 and Probability 2.

A review of the essentials can also be found in the supplementary Probability and Distribution Refresher notes. These supplementary notes also provide details of probability distributions frequently employed in statistical analysis.

2  Distributions for statistical models briefly revisits the distributions relevant for this module.

Statistics

For a refresher of statistical concepts discussed in earlier statistics courses and required in this module see Appendix A — Statistics refresher.