Module overview
The module is designed for postgraduate students whose first degree is in Mathematics or another discipline where development of mathematical skills is a significant component (Science, Engineering, Economics, Quantitative Social Sciences). While the material covered is similar in technical level to that which might be found in an undergraduate mathematics curriculum, the quantity of material is much larger, and the pace of delivery correspondingly much faster. Hence the module requires students to have developed study skills to graduate level.
The module is comprised of three submodules, in probability and distribution theory, statistical inference and statistical modelling, as described in the syllabus below.
Aims and Objectives
Learning Outcomes
Learning Outcomes
Having successfully completed this module you will be able to:
- Explain the concepts of a compound distribution, and apply them.
- Recall the principles of statistical model selection, and apply a range of different approaches to model selection
- Explain the concepts of random variable, probability distribution, distribution function, expected value, variance and higher moments, and calculate expected values and probabilities associated with the distributions of random variables
- Define a probability generating function, a moment generating function, a cumulant generating function and cumulants, derive them in simple cases, and use them to evaluate moments
- Describe the main methods of estimation and the main properties of estimators, and apply them
- Define basic discrete and continuous distributions, be able to apply them and simulate them in simple cases
- Explain the principles of Bayesian statistical inference
- Test statistical hypotheses.
- Analyse the dependence of a binary response variable on explanatory variables using logistic regression models
- Construct confidence intervals for unknown parameters.
- Investigate relationships between variables using regression models.
- Explain the concepts of independence, jointly distributed random variables and conditional distributions, and use generating functions to establish the distribution of linear combinations of independent random variables.
- Summarise the main features of a data set (exploratory data analysis).
- Explain the concepts of random sampling, statistical inference and sampling distribution, and state and use basic sampling distributions.
- Use appropriate software (say R) to fit a multiple linear regression model to a data set and interpret the output.
- Explain the concepts of probability, including conditional probability
- Explain the concepts of analysis of variance and use them to investigate factorial dependence
- State the central limit theorem, and apply it.
- Explain the concept of likelihood and derive the likelihood and associated functions of interest for simple models
- Recall the definition of a generalised linear model
Syllabus
Probability theory:
- Basic concepts; Axioms; Addition laws; Independence
- Random variables and probability distributions; Expectation, moments and variance;
- Distribution function
- Discrete and continuous distributions; Calculating probabilities and expectations;
- Transformations of random variables
- Examples: binomial, Poisson, exponential, normal etc.
- Generating functions; Properties and applications; Cumulants
- Joint distributions; Independence; Covariance and correlation
- Conditional probability; Bayes theorem; Compound distributions
Statistical inference:
- Sampling concepts; Samples and populations; Sampling distribution
- The central limit theorem; normal approximations
- Point estimation; Efficiency; Bias; Consistency; Mean squared error; Method of Moments
- Confidence intervals; Normal, Poisson and binomial examples
- Hypothesis testing; Terminology; Normal, Poisson and binomial examples; Goodness-of-fit tests
- Likelihood; Maximum likelihood estimation and its asymptotic properties
- Introduction to Bayesian ideas; Prior and posterior distributions
Statistical modelling:
- Summarising data; Summary statistics; Simple graphical displays
- Regression and linear models; Least squares estimation; Inference for regression coefficients;
- Comparing models; Multiple regression; Residuals and model criticism; Prediction
- Analysis of variance
- Use appropriate software (say R) to fit a multiple linear regression model to a data set and interpret the output.
- Introduction to generalised linear models; Exponential family; Logistic regression
- Concepts and methods of statistical model selection; Hypothesis testing approaches;
- Information criteria.
Learning and Teaching
Teaching and learning methods
Lectures, assigned problems, private study
Type | Hours |
---|---|
Teaching | 36 |
Independent Study | 114 |
Total study time | 150 |
Resources & Reading list
Textbooks
Mendenhall W, Wackerly DD and Scheaffer RL (2007). Mathematical Statistics with Applications. Brooks/Cole.
Larsen RJ and Marx ML (2005). An Introduction to Mathematical Statistics and Its Applications. Pearson.
Assessment
Summative
This is how we’ll formally assess what you have learned in this module.
Method | Percentage contribution |
---|---|
Class Test | 10% |
Exam | 70% |
Coursework | 20% |
Referral
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Method | Percentage contribution |
---|---|
Exam | 100% |
Repeat Information
Repeat type: Internal & External