**Course co-ordinator(s):** Dr James R Cruise (Edinburgh), Dr Soo Huei Ching (Malaysia).

**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, 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.

## Detailed Information

**Pre-requisite course(s):** F77SA Introduction to Statistical Science A & F77SB Introduction to Statistical Science B .

**Location: **Edinburgh, Malaysia.

**Semester: **1.

**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, 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

**Reading list:**

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 Methods:**

2 hour end-of-course examination | (85% weight) |

continuous assessment | (15% weight) |

**SCQF Level: **8.

## 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