*MATH6122 *Probability and Mathematical Statistics

## 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 devloped 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

#### Module Aims

To provide students with a solid grounding in probability theory and mathematical statistics.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

- Explain the concepts of probability, including conditional probability
- Explain the concept of likelihood and derive the likelihood and associated functions of interest for simple models
- Construct confidence intervals for unknown parameters.
- Test statistical hypotheses.
- Explain the principles of Bayesian statistical inference
- Summarise the main features of a data set (exploratory data analysis).
- Investigate relationships between variables using regression models.
- Explain the concepts of analysis of variance and use them to investigate factorial dependence
- Recall the definition of a generalised linear model
- Analyse the dependence of a binary response variable on explanatory variables using logistic regression models
- 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
- Define basic discrete and continuous distributions, be able to apply them and simulate them in simple cases
- 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.
- Explain the concepts of a compound distribution, and apply them.
- State the central limit theorem, and apply it.
- Explain the concepts of random sampling, statistical inference and sampling distribution, and state and use basic sampling distributions.
- Describe the main methods of estimation and the main properties of estimators, and apply them

### 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 • 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

Mendenhall W, Wackerly DD and Scheaffer RL (2007). Mathematical Statistics with Applications.

Larsen RJ and Marx ML (2005). An Introduction to Mathematical Statistics and Its Applications.

### Assessment

#### Summative

Method | Percentage contribution |
---|---|

Class Test (35 minutes) | 10% |

Exam (3 hours) | 90% |

#### Referral

Method | Percentage contribution |
---|---|

Exam (3 hours) | 100% |

#### Repeat Information

**Repeat type: Internal & External**

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.