**Lecturer:** Roger Gray

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.

Together with second-term course F79SN Further Statistical Methods, this course covers the material of subject CT3 of the UK actuarial profession.

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

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.

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 Ed.), PWS-Kent.

3 hour examination (combined with Semester 2 course) for MSc in Actuarial Science; 2 hour examination for all other students.

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

- 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

Information for Current Students

Staff Information

Timetables & Programme Structures

Support Services

**Actuarial Mathematics and Statistics,**

School of Mathematical and Computer Sciences ,

Heriot-Watt University,
Edinburgh
EH14 4AS, Scotland

Phone: +44 (0)131 451 3202, Fax +44 (0)131 451 3327 or Email enquiries@macs.hw.ac.uk