Prerequisites

Statistics is a mathematical science that requires practical working knowledge of vector and matrices and calculus of functions of several variables, as well as an deeper understanding of probability and distributions. Furthermore, as Year 2 Semester 2 course the MATH27720 Statistics 2 module builds on previous statistics modules.

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

Probability and distributions

For an overview of essential concepts in probability and of probability distributions frequently employed in statistical analysis please refer to the supplementary Probability and Distribution Refresher notes.

Specifically, in this module we will make use of the distributions listed below.

Univariate distributions

  • Bernoulli distribution \(\text{Ber}(\theta)\)

  • Binomial distribution \(\text{Bin}(n, \theta)\)

  • Normal distribution \(N(\mu, \sigma^2)\)

  • Gamma distribution \(\text{Gam}(\alpha, \theta)\)
    Other names and parameterisations for the gamma distribution are:

    • univariate Wishart distribution \(W_1\left(s^2, k \right)\)
    • scaled chi-squared distribution \(s^2 \text{$\chi^2_{k}$}\)

    Special cases of the gamma distribution are:

    • chi-squared distribution \(\text{$\chi^2_{k}$}\)
    • exponential distribution \(\text{Exp}(\theta)\)

  • Location-scale \(t\)-distribution \(\text{lst}(\mu, \tau^2, \nu)\) Special case of the location-scale \(t\)-distribution are:

    • Student’s \(t\)-distribution \(t_\nu\)
    • Cauchy distribution \(\text{Cau}(\mu, \tau)\)

Further univariate distributions used in Bayesian analysis

  • Beta distribution \(\text{Beta}(\alpha, \beta)\) (prior for a proportion)
  • Inverse gamma distribution \(\text{Inv-Gam}(\alpha, \beta)\) (prior for a variance)

Multivariate distributions

  • Categorical distribution \(\text{Cat}(\symbfit \pi)\)
  • Multinomial distribution \(\text{Mult}(n, \symbfit \pi)\)
  • Multivariate normal distribution \(N_d(\symbfit \mu, \symbfit \Sigma)\)

Statistics

For a refresher of some essential concepts in statistics see Appendix A — Statistics refresher of these notes.