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.
Reading
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
