Statistics (Part IB, Lent 2025)

General information

• Please email me or leave a comment below if you find any mistakes in the lecture slides/notes.

• Lectures will be recorded and the recordings can be found on Moodle.

Lectures notes

Course materials will be continuously updated.

• Slides in lectures.
• LaTex notes (exactly the same as the slides but in a more printer-friendly format).
• Handwritten notes in the lectures.
• R code for some demonstrations in the lectures.
  • To execute this code, you need to first download and install R from CRAN.
  • You may find it useful to use an integrated development environment (DE) such as RStudio Desktop or Emacs Speaks Statistics.
  • Use of statistical software is NOT examinable.

Further notes can be found on David Spiegelhalter’s webpage.

Example sheets

• Example sheets can be found on the Faculty website.

• Common probability distributions.

Schedules

Estimation

Review of distribution and density functions, parametric families. Examples: binomial, Poisson, gamma. Sufficiency, minimal sufficiency, the Rao–Blackwell theorem. Maximum likelihood estimation. Confidence intervals. Use of prior distributions and Bayesian inference. [6]

Hypothesis testing

Simple examples of hypothesis testing, null and alternative hypothesis, critical region, size, power, type I and type II errors, Neyman–Pearson lemma. Significance level of outcome. Uniformly most powerful tests. Likelihood ratio, and use of generalised likelihood ratio to construct test statistics for composite hypotheses. Examples, including t-tests and F -tests. Relationship with confidence intervals. Goodness-of-fit tests and contingency tables. [4]

Linear models

Derivation and joint distribution of maximum likelihood estimators, least squares, Gauss-Markov theorem. Testing hypotheses, geometric interpretation. Examples, including simple linear regression and one-way analysis of variance. ∗Use of software∗. [6]

Appropriate books

G. Casella and R.L. Berger. Statistical Inference. Duxbury 2001

D.A. Berry and B.W. Lindgren. Statistics, Theory and Methods. Wadsworth 1995

M.H. DeGroot and M.J. Schervish Probability and Statistics. Pearson Education 2001