# School of Mathematical and Computer Sciences

## F78PA Probability and Statistics A

Lecturer: Dr. Jennie Hansen

#### Aims

The aims of this course are

• To develop the tools of probability theory with a view to applications in statistical inference and actuarial science
• To provide an introduction to computer simulation in R and its applictions to probability and statistics.

#### Summary

In this course we develop probability models for random phenomena. In particular, we develop the methodology needed for the study of random variables and their distributions. Random variables are essential to the modelling of most random phenomena, and have applications in statistical science, financial mathematics, and actuarial science. Common discrete and continuous random variables (Bernoulli, binomial, geometric, hypergeometric, Poisson, uniform, normal, lognormal, exponential, gamma) which are frequently used for modelling are introduced and their properties investigated. We also introduce multivariate distributions, conditional distributions, and criteria for independence of random variables. We study sums of independent random variables, and introduce the weak law of large numbers and the central limit theorem.

We will use computer simulation as an aid to understanding the behaviour of probabilistic and statistical models, and to doing calculations for these models.

Some recommended textbooks are:

• D. Stirzaker (1999), Probability and Random Variables: a beginner's guide, Cambridge University Press.
• G. Grimmett & D. Welsh (1990), Probability: an Introduction, Oxford University Press.
• S. M. Ross (2006), A First Course in Probability, 7th edition, Pearson.

#### Assessment

 2 hour end-of-course examination (85% weight) continuous assessment (15% weight)

#### 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 will be available on Vision

#### Detailed syllabus

• Probability models: sample spaces, events, probability measures, axioms and properties
• Random variables and their distributions: distribution, probability and density functions, transformations of random variables
• Expectation, variance, and standard deviation of random variables
• Important special distributions and their main properties: Bernoulli, Binomial, Geometric, Hypergeometric, Poisson, Uniform, Normal, Lognormal, Exponential, Gamma
• Conditional probability and independence: including chain rule, partition rule, Bayes' Theorem
• Joint probability, density and distribution functions
• Marginal and conditional distributions
• Independent random variables and sums of independent random variables
• Generating functions and their applications
• Markov and Chebychev inequalities, the weak law of large numbers, and the Central Limit Theorem, with applications to statistics
• Expectation of a function of random variables, covariance, correlation
• Conditional expectation and its uses
• Computer simulation and its applications in probability and statistics
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