Undergraduate Programme and Module Handbook 2024-2025

# Module MATH1597: Probability I

## Department: Mathematical Sciences

### MATH1597: Probability I

Type | Open | Level | 1 | Credits | 10 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- A level Mathematics at grade A or better and AS level Further Mathematics at grade A or better, or equivalent.

#### Corequisites

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

#### Excluded Combination of Modules

- Mathematics for Engineers and Scientists (MATH1551), Single Mathematics A (MATH1561), Single Mathematics B (MATH1571) may not be taken with or after this module.

#### Aims

- 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.

#### Content

- 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.

#### Learning Outcomes

- 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.

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

- Problem solving.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- 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.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 27 | 3 pw in wks 1-3, 5, 7, 9; 2 pw in wks 4, 6, 8, 10; 1 revision in wk 21 | 1 Hour | 27 | |

Tutorials | 6 | 1 per week in weeks 2, 4, 6, 8, 10 (Term 1), plus 1 revision tutorial in Term 3 | 1 Hour | 6 | ■ |

Problems Classes | 4 | 1 pw in wks 4, 6, 8, 10 | 1 Hour | 4 | |

Support classes | 9 | 1 pw in wks 2-10 | 1 Hour | 9 | |

Preparation and Reading | 54 | ||||

Total | 100 |

#### Summative Assessment

Component: Continuous Assessment | Component Weighting: 10% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

4 summative assessment assignments during Term 1 | 100% | ||

Component: Examination | Component Weighting: 90% | ||

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 2 hours | 100% | Yes |

#### Formative Assessment:

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.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University