# School of Mathematical and Computer Sciences

## F71SM Statistical Methods

Lecturer: Roger Gray

#### 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.

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

#### Summary

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

#### Learning outcomes

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.

#### Assessment

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

#### Help

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

#### Course web page

Further information and course materials are available at http://www.ma.hw.ac.uk/~roger/F71SM1/

#### Detailed 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 (χ2)
• sampling distribution of a proportion
• ratio of 2 sample variances (F-distribution)
• 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 μ12
• CIs for ratio of 2 population variances σ1222
• 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 μ12
• equality of 2 population variances σ1222
• Linear regression
• response and explanatory variables
• linear regression model
• least squares estimation
• sums of squares, coefficient of determination R2
• inference on regression parameters and tests for significance of regression
• predicting a mean response and an actual response
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