MA1500: Introduction to Probability Theory

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA1500
External Subject Code 101032
Number of Credits 10
Level L4
Language of Delivery English
Module Leader Dr Andreas Artemiou
Semester Autumn Semester
Academic Year 2018/9

Outline Description of Module

The module begins with the idea of a probability space, which is how we model the possible outcomes of a random experiment. Concepts such as statistical independence and conditional probability are introduced, and a number of practical problems are studied. We then turn our attention to random variables, and look at some well-known probability distributions. Following this we focus on discrete distributions, and introduce the idea of independence for random variables, and the important concept of mathematical expectation. This leads on to the study of random vectors, where we introduce covariance and correlation, conditional distributions and the law of total expectation. Finally, we show how the ideas developed for discrete distributions can be carried over to continuous distributions, and conclude with some approximation theorems.  

This is a lecture-based module. Students will be required to demonstrate problem-solving skills throughout the module. No previous knowledge of probability theory is assumed.

The module is intended to prepare students for subsequent modules involving probability and statistics within the degree scheme.

Free Standing Module Requirements:  A pass in A-Level Mathematics of at least Grade A

On completion of the module a student should be able to

  • Appreciate the theoretical foundations of probability theory.
  • Apply the concepts of independence and conditional probability to practical problems.
  • Understand the concepts of random variables and distribution functions.
  • Derive the expected value and variance of several well-known distributions (including the binomial, Poisson, geometric, exponential and normal distributions)
  • Apply the concepts of independent random variables and conditional expectation to practical problems.
  • Apply various approximation theorems to practical problems.

How the module will be delivered

27 fifty-minute lectures

5 fifty-minute tutorial classes

This is a lecture based module. Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of worked solutions for tutorial classes.

Skills that will be practised and developed

Skills:

To choose appropriate ideas and techniques to solve practical problems that involve randomness.

Transferable Skills:

Formulating problems and presenting clear solutions through logical reasoning.

How the module will be assessed

Formative assessment is carried out by means of regular tutorial exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during tutorial classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short.  Section B has a choice of two from three equally weighted questions.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Introduction To Probability Theory 2

Syllabus content

  • Set algebra.
  • Probability spaces.
  • Conditional probability and Bayes’ theorem
  • Independent events.
  • Random variables and distribution functions.
  • Discrete random variables.
  • Mathematical expectation.
  • Covariance and correlation.
  • Conditional distributions and conditional expectation.
  • Continuous random variables.
  • Medians, quantiles and statistical tables.
  • Approximation to the binomial and Poisson distributions.

Background Reading and Resource List

Grimmet G., & Stirzaker D. 2001. Probability and Random Processes. Oxford University Press.


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