Prerequisites

Statistics is a mathematical science that requires practical working knowledge not only in probability 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 for a real symmetric matrix
  • Positive and negative definiteness of a real symmetric matrix (containing 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 \(\log(x)\), when the base is not without explicitly stated, always refer to the natural logarithm. For logarithms with respect to base 2 and 10 we denote them as \(\log_2(x)\) and \(\log_{10}(x)\), respectively.

Combinatorics, probability and distributions

For a detailed introduction of concepts in combinatorics 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.