**Course co-ordinator(s):** Prof George Streftaris (Edinburgh).

**Aims:**

This course aims to provide postgraduate students taking the MSc in Actuarial Science, the MSc in Financial Mathematics, and other courses with a broad knowledge of the principal areas of mathematical statistics and statistical methods widely used in insurance and finance.

This course partially covers the material of CT3.

**Summary:**

- Data summary
- Basic probability concepts
- Random variables and their distributions
- Joint distributions
- Central limit theorm
- Parameter estimation
- Statistical inference
- Linear regression

## Detailed Information

**Course Description: **Link to Official Course Descriptor.

**Pre-requisites:** none.

**Location: **Edinburgh.

**Semester: **1.

**Syllabus:**

- Summarising and displaying data

- Probability and random variables
- Random experiments, sample spaces, events
- Probability axioms
- Conditional probability
- Independent events
- Random variables
- Density and distribution functions
- Expected values
- Moments and generating functions
- Functions of a random variable
- Some special discrete distributions
- uniform
- bernoulli
- binomial
- geometric
- negative binomial
- hypergeometric
- Poisson

- Some special continuous distributions
- uniform
- exponential, gamma
- normal, chi-square

- Joint distribution of several random variables
- joint, marginal distributions
- conditional distributions

- Conditional expectation
- Markov and Chebyshev (Tchebyshev) inequality, laws of large numbers
- Central limit theorem
- Sampling distributions
- sampling distribution of the mean (normal,
*t*-distribution) - sampling distribution of the variance (χ
^{2}) - sampling distribution of a proportion
- ratio of 2 sample variances (
*F*-distribution)

- sampling distribution of the mean (normal,

- Statistical inference
- Estimation
- by method of moments
- by maximum likelihood
- Properties of estimators
- unbiasedness
- efficiency
- Cramèr-Rao lower bound
- consistency

- Confidence intervals: definition and construction
- CIs for population mean and variance
- CI for Poisson mean λ
- CI for a proportion
- CIs for difference between 2 population means μ
_{1}-μ_{2} - CIs for ratio of 2 population variances σ
_{1}^{2}/σ_{2}^{2}

- Hypothesis testing
- null and alternative hypotheses
- test statistics and relation to confidence interval pivotal quantities
- decision errors, significance level,
*p*-values - power of tests
- likelihood ratio
- tests for
- population mean and variance
- equality of 2 populations means μ
_{1}-μ_{2} - equality of 2 population variances σ
_{1}^{2}/σ_{2}^{2}

- Estimation
- Linear regression
- response and explanatory variables
- linear regression model
- least squares estimation
- sums of squares, coefficient of determination
*R*^{2} - inference on regression parameters and tests for significance of regression
- predicting a mean response and an actual response

**Learning Outcomes: Subject Mastery**

At the end of studying this course, students should be able to:

- Summarise and display data.
- Perform basic probability calculations.
- Calculate moments and the expected values of other functions of random variables.
- Apply the central limit theorem.
- Obtain estimators of parameters of certain common distributions.
- Determine properties of estimators: efficiency, Cramèr-Rao lower bound, (approx. large-sample) distribution.
- Perform inference on parameter estimates: obtain confidence intervals and carry out hypothesis testing.
- Fit a linear regression model.

**Reading list:**

The required sets of tables (provided) are:

- D V Lindley & W F Scott:
*New Cambridge Statistical Tables*, Second edition, CUP 1995. *Formulae and Tables for Examinations of the The Faculty of Actuaries and the Institute of Actuaries*, 2002

Some students have found the following books helpful. The first book (Miller and Miller) is the required text-book. The second book (Rees) is an elementary introduction to some topics and is recommended for students with little or no previous study of statistics.

- Miller and Miller:
*John E. Freund’s Mathematical Statistics with Applications*(7th or later edition), Pearson Prentice-Hall. - Rees:
*Essential Statistics*(4th or later edition) Chapman and Hall/CRC - H. J. Larson:
*Introduction to Probability Theory and Statistical Inference*(3rd Ed.), Wiley. - W. Mendenhall and R. J. Beaver:
*Introduction to Probability and Statistics*(8th or later edition.), Brooks/Cole.

**Assessment Methods:**

Examination will be at least 60% and no more than 80%.

Coursework will be at least 20% and no more than 40%.

Re-assessment in the next academic year

**SCQF Level: **11.

## Other Information

**Help:** If you have any problems or questions
regarding the course, you are encouraged to contact the lecturer

**VISION:** further information and course materials
are available on VISION