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MA1061 Probability

MA1061 Probability

Credits: 10 Convenor: Dr. M. Tretyakov Semester: 2 (weeks 15 to 26)
Prerequisites: desirable: MA1001,MA1102
Assessment: Coursework: 20% One and a half hour examination: 80%
Lectures: 18 Problem Classes: 5
Tutorials: none Private Study: 52
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Subject Knowledge

Aims

To develop a strong intuitive understanding of the basic ideas of probability theory, probabilistic models, random variables and their distributions.

Learning Outcomes

On completion of this module, students should:
$\bullet$
understand the concept of a probabilistic model for an experiment with finite number of outcomes (a finite probability space); sample space, algebra of events, probability and its properties;
$\bullet$
be able to compute probabilities for events defined on a sample space of equally likely outcomes;
$\bullet$
understand the concept of conditional probability and independence;
$\bullet$
understand the concept of random variable and the differences between discrete and continuous random variables;
$\bullet$
know what is meant by a probability distribution or probability density function; be able to apply the appropriate methods for computing probabilities, expectations and variances;
$\bullet$
understand the genesis of the binomial, geometric and Poisson distributions;
$\bullet$
be aware of the normal distribution, its ubiquity, its parameters and their interpretation, the general shape of its probability density function, and finding probabilities using standard tables;
$\bullet$
know the basic properties of the mean and variance of random variables;
$\bullet$
understand the content and consequences of the DeMoivre-Laplace and Central Limit Theorems and be able to apply them to simple problems.

Methods

Class sessions together with some handouts.

Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 4 questions with full marks on the paper obtainable from 3 complete answers.

Subject Skills

Aims

Develop problem solving skills, written communication skills.

Learning Outcomes

Students will have to investigate problems, draw conclusions and make conjectures. Students will be able to use the techniques taught within the module to solve problems, and be able to present arguments and solutions in a coherent and logical form. An understanding of the concept of probability and the interpretation of simple probability statements. A reasonable grasp of the material covered in this module forms an essential prerequisite for all later courses in probability and statistics, or indeed any discipline employing probabilistic modelling.

Methods

Class sessions, coursework, exam.

Assessment

Marked problem sheets, exam.

Explanation of Pre-requisites

There are few formal prerequisites beyond what is normally covered in $A$-level mathematics syllabuses. The module MA1001 provides the proofs of some calculus results which are used in this module, and introduces some basic multivariate calculus techniques. The module MA1102 provides some basics of the elementary set theory.

Course Description

Probability statements are almost unavoidable, and probabilistic models pervade most areas of science. This course introduces the basic ideas and rules of probability, together with some simple probabilistic models and techniques for computing the probabilities of events. We introduce the important concept of conditional probabilities and independence. We also introduce random variable together with its probability distribution, expectation and variance. A number of important distributions are considered, including the binomial, geometric, Poisson and normal distributions. The DeMoivre-Laplace and Central Limit Theorem are given without proofs.

Syllabus

  1. Probabilistic model with finite number of outcomes (sample space, algebra of events, probability); properties of probability; finite probability spaces with equally likely outcomes.
  2. Conditional probability and independence; properties of conditional probability.
  3. Discrete random variables and their characteristics; probability distribution, expectation, variance; Bernoulli distribution; Bernoulli trials and binomial distribution.
  4. Limit theorems (the DeMoivre-Laplace and Central Limit Theorems). Poisson distribution; geometric distribution.
  5. Continuous distributions; normal (Gaussian) distribution.

Reading list

Recommended:

M. H. DeGroot, Probability and Statistics, 2nd edition, Addison-Wesley, 1986. W. Mendenhall, R. L. Scheaffer and D. D. Wackerly, Mathematical Statistics with Applications, 4th edition, Duxbury Press, 1990. P. L. Meyer, Introductory probability and statistical applications, Addison-Wesley, 1970. S. Lipschutz and M. Lipson, Probability, Shaum's outlines, McGraw-Hill, 2000.

Resources

Problem sheets, additional handouts, lecture rooms.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1101 Proof and Logical Structures Up: ModuleGuide03-04 Previous: MA1051 Newtonian Dynamics

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Last updated: 2004-02-21
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