MATH 2274 Probability Theory I
This is an introductory course that approaches probability theory from two perspectives: Probability theory is a branch of mathematics. As such, we will focus on the fundamental assumptions of Probability Theory and how the main properties of Probability Measures proceed from these assumptions.
Throughout the course, therefore, students will be expected to be able to derive the main results that they use. Very little will be assumed without proof. Probability Theory is primarily concerned with modelling phenomena with uncertain outcomes. The course emphasizes this. It is most definitely not a course in Pure Mathematics. A knowledge of calculus (including a good understanding of limits, continuity, differentiability) is assumed (hence the need for Math1150). An appreciation of the idea of proof is expected but Math1140 is not essential (though it is desirable). The course begins with a discussion of the basic ideas of probability, including the axioms of probability, combinatorial probability, conditional probability and independence. The rest of the course focuses on distribution theory. The distribution theory of one discrete and one continuous random variable is discussed. Special attention is paid to well-known discrete and continuous distributions such as the Bernoulli, Binomial, Poisson, Exponential, Gamma and Normal. Then the distribution theory of several random variables is discussed. The idea of a statistic is introduced and the distribution theory of the mean and the sample variance is described. This leads finally to the idea of convergence in distribution and the Central Limit Theorem (without proof) The approach taken is non-rigorous. In particular, there will be no mention of sigma algebras or of measure theory. Assessment is designed to encourage students to work continuously with the course materials. Active learning will be achieved through weekly assignments and problem sheets allowing continuous feedback and guidance on problem solving techniques in tutorials and lectures.