MATH1597: Probability I

• A level Mathematics at grade A or better and ASlevel Further Mathematics at grade A or better, orequivalent.

• Calculus I (Maths Hons) (MATH1081) or Calculus I (MATH1061)

• Mathematics for Engineers and Scientists (MATH1551), SingleMathematics A (MATH1561), Single Mathematics B (MATH1571) may not be taken with or after thismodule.

• This module will give an introduction to the mathematics of probability.
• It will present a mathematical subject of key importance to the real-world ("applied") that is nevertheless based on rigorous mathematical foundations ("pure").
• It will present students with a wide range of mathematical ideas in preparation for more demanding and specialized material later.
• A range of topics are treated each at an elementary level to give a foundation of basic concepts, results, and computational techniques.
• A rigorous approach is expected.

• Introduction to probability: chance experiments, sample spaces, events, assigning probabilities. Probability axioms and interpretations.
• Conditional probability: theorem of total probability, Bayes's theorem. Independence of events.
• Applications of probability: reliability networks, genetics.
• Random variables: discrete probability distributions and distribution functions, binomial and Poisson distributions, Poisson approximation to binomial, transformations of random variables.
• Continuous random variables: probability density functions, uniform, exponential, and normal distributions.
• Joint, marginal and conditional distributions. Independence of random variables.
• Expectations: expectation of transformations, variance, covariance, conditional expectation, partition theorem for expectations, Markov and Chebyshev inequalities.
• Limit theorems: Weak law of large numbers, central limit theorem, moment generating functions.

Subject-specific Knowledge:

• By the end of the module students will:
• be able to solve a range of both routine and more challenging problems in probability theory.
• be familiar with the basic mathematical concepts of probability theory.
• have a broad knowledge of the subject area demonstrated by detailed familiarity with the following topics:
• set theoretic framework for sample spaces and events, including notions of countable and uncountable sets;
• event calculus, probability axioms, conditional probability, Bayes's formula, independence of events;
• discrete and continuous random variables and their distributions, including particular familiarity with the binomial, Poisson, normal and exponential distributions;
• joint distributions, conditional distributions, and independence of random variables;
• expected value of a random variable, variance, covariance, and moment generating functions;
• tail inequalities, the weak law of large numbers, and the central limit theorem.

Subject-specific Skills:

• Students will have basic mathematical skills in the following areas: modelling, abstract reasoning, numeracy.

Key Skills:

• Problem solving.

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Tutorials provide active engagement and feedback to the learning process.
• Students are expected to develop their knowledge and skills with at least 50 hours of self-study.
• Homework problems provide formative assessment to guide students in the correct development of their knowledge and skills. They are also an aid in developing students' awareness of standards required.
• Initial diagnostic testing and associated supplementary problems classes fill in gaps related to the wide variety of syllabuses available at Mathematics A-level.
• The end-of-year written examination provides a rigorous assessment of the mastery of the lecture material.

Term 1: Fortnightly written or electronic assignments to be assessed and returned. Other assignments are set for self-study and solutions are made available to students. Students will have about one week to complete each assignment. Term 2: 45 minute collection paper in the beginning of Epiphany term.

If you have a question about Durham's modular degree programmes, please visit our FAQ webpages, Help page or our glossary of terms. If you have a question about modular programmes that is not covered by the FAQ, or a query about the on-line Undergraduate Module Handbook, please contact us.

Prospective Students: If you have a query about a specific module or degree programme, please Ask Us.

Current Students: Please contact your department.