ST115 Introduction to Probability
Please note that all lectures for Statistics modules taught in the 2022-23 academic year will be delivered on campus, and that the information below relates only to the hybrid teaching methods utilised in 2021-22 as a response to Coronavirus. We will update the Additional Information (linked on the right side of this page) prior to the start of the 2022/23 academic year.
Throughout the 2021-2022 academic year, we will be adapting the way we teach and assess your modules in line with government guidance on social distancing and other protective measures in response to Coronavirus. Teaching will vary between online and on-campus delivery through the year, and you should read the additional information linked on the right hand side of this page for details of how this will work for this module. The contact hours shown in the module information below are superseded by the additional information. You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.
All dates for assessments for Statistics modules, including coursework and examinations, can be found in the Statistics Assessment Handbook at http://go.warwick.ac.uk/STassessmenthandbook
ST115-12 Introduction to Probability
Introductory description
The module runs in Term 2 and provides elementary introduction to the theory of probability. The topics include axioms of probability, combinatorics, independent events, conditional probability, random variables, discrete and continuous probability distributions, expectation and variance, joint probability distributions, independence of random variables, sum of independent random variables, covariance and correlation.
This module is core for students with their home department in Statistics and is not available to students from other departments. Students from other departments should consider ST111 Probability A and ST112 Probability B instead.
Module aims
To lay the foundation for all subsequent modules in probability and statistics, by introducing the key notions of mathematical probability and developing the techniques for working with probability distributions and random variables.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
- Experiments with random outcomes: the notions of random experiment, sample space and events. Operations with sets and their interpretation.
- Axioms of probability. Properties of a probability measure: Boole’s inequality, continuity of a probability measure, inclusion-exclusion formula.
- Finite sample spaces with equally likely outcomes.
- Independence of events. Conditional probabilities. Bayes theorem.
- The notion of a random variable. Examples in both discrete and continuous settings. Indicator random variables.
- The notion of the distribution of a random variable. Probability mass functions and density functions. Cumulative distribution functions.
- Expectation of random variables. Properties of expectation.
- Mean and variance of distributions. Chebyshev's inequality.
- Independence of random variables. Joint distributions. Covariance and correlation. Cauchy-Schwartz inequality.
- Addition of independent random variables: convolutions. Moment generating function and use to compute convolutions.
- Important families of distributions: Binomial, Poisson, negative Binomial, exponential, Gamma and Gaussian. Their properties, genesis and inter-relationships.
Learning outcomes
By the end of the module, students should be able to:
- Understand key notions of mathematical probability including random variables and their distributions
- Appreciate the role of randomness in mathematical modelling of real world situations.
- Use appropriate mathematical techniques to calculate the probabilities of events, and the expectations of random variables
Indicative reading list
Ross, A first course in probability, Prentice Hall, 1994
Pitman, Probability, Springer texts in Statistics
Suhov and Kelbert, Probability and Statistics by Example: Basic Probability and Statistics.
View reading list on Talis Aspire
Subject specific skills
Mathematical, analytical, problem solving
Transferable skills
Analytical, problem solving, investigative skills, communication, good working habits.
Study time
| Type | Required | Optional |
|---|---|---|
| Lectures | 30 sessions of 1 hour (26%) | 2 sessions of 1 hour |
| Seminars | 8 sessions of 1 hour (7%) | |
| Tutorials | 5 sessions of 1 hour (4%) | |
| Private study | 53 hours (46%) | |
| Assessment | 18 hours (16%) | |
| Total | 114 hours |
Private study description
Weekly revision of lecture notes and materials, wider reading and practice exercises, working on problem sets and preparing for examination.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Students can register for this module without taking any assessment.
Assessment group D2
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| Multiple Choice Quiz 1 | 2% | 3 hours | Yes (waive) |
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A multiple choice quiz which will take place during the term that the module is delivered. |
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| Problem set 1 | 5% | 6 hours | Yes (extension) |
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A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a well-prepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. |
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| Multiple Choice Quiz 2 | 3% | 3 hours | Yes (waive) |
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A multiple choice quiz which will take place during the term that the module is delivered. |
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| Problem set 2 | 5% | 6 hours | Yes (extension) |
|
A problem sheet that include problem solving and calculations. Problem sheets will be set at fortnightly intervals. The problem sheets will contain a number of questions for which solutions and / or written responses will be required. The preparation and completion time noted below refers to the amount of time in hours that a well-prepared student who has attended lectures and carried out an appropriate amount of independent study on the material could expect to spend on this assignment. |
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| In-person Examination | 85% | No | |
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The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. ~Platforms - Moodle
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Assessment group R1
| Weighting | Study time | Eligible for self-certification | |
|---|---|---|---|
| In-person Examination - Resit | 100% | No | |
|
The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
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Feedback on assessment
Answers to problems sets will be marked and returned to students in a tutorial or seminar taking place the following week when students will have the opportunity to discuss it.
Solutions and cohort level feedback will be provided for the examination.
Courses
This module is Core for:
- Year 1 of USTA-G302 Undergraduate Data Science
- Year 1 of USTA-G304 Undergraduate Data Science (MSci)
- Year 1 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
- Year 1 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
- Year 1 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)
- Year 1 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics