# ST115-12 Introduction to Probability

21/22
Department
Statistics
Level
Giuseppe Cannizzaro
Credit value
12
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry

##### 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.

1. Experiments with random outcomes: the notions of random experiment, sample space and events. Operations with sets and their interpretation.
2. Axioms of probability. Properties of a probability measure: Boole’s inequality, continuity of a probability measure, inclusion-exclusion formula.
3. Finite sample spaces with equally likely outcomes.
4. Independence of events. Conditional probabilities. Bayes theorem.
5. The notion of a random variable. Examples in both discrete and continuous settings. Indicator random variables.
6. The notion of the distribution of a random variable. Probability mass functions and density functions. Cumulative distribution functions.
7. Expectation of random variables. Properties of expectation.
8. Mean and variance of distributions. Chebyshev's inequality.
9. Independence of random variables. Joint distributions. Covariance and correlation. Cauchy-Schwartz inequality.
10. Addition of independent random variables: convolutions. Moment generating function and use to compute convolutions.
11. 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

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.

##### 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
Multiple Choice Quiz 1 2% 3 hours

A multiple choice quiz which will take place during the term that the module is delivered.

Problem set 1 5% 6 hours

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.

Multiple Choice Quiz 2 3% 3 hours

A multiple choice quiz which will take place during the term that the module is delivered.

Problem set 2 5% 6 hours

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.

In-person Examination 85%

The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
Calculators are NOT permitted in this examination.

~Platforms - Moodle

• Cambridge Statistical Tables (blue)
• Students may use a calculator
##### Assessment group R1
Weighting Study time
In-person Examination - Resit 100%

The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade.
Calculators are NOT permitted in this examination.

• Cambridge Statistical Tables (blue)
• Students may use a calculator
##### 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

This module ended in 2020/21 and this information is only relevant to students with reassessments.