Topic outline

  • General

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  • Preparing for the exam

  • Module description

    This module further develops the ideas introduced in the first year Probability and Statistics module. Probability is about quantifying uncertainty; we typically have a situation or system with some randomness to it and want to determine numerically the chance of it behaving in a certain way. Statistics on the other hand involves the collection, organization, analysis, interpretation and presentation of data; often we have a (perhaps very large) collection of data and want to deduce something about the process or population behind it.

    We will learn about more advanced theoretical notions of probability, such as the distributions of random variables, their independence, their transformations and limit theorems. We will then look at different types of statistical tests of hypotheses. By the end of this module you will be able to addresses the questions of when and how to use them in real problems from life and physical sciences, business and economics, psychology and many other areas.


  • Syllabus

    • See the table of content for lecture notes.

  • Module aims and learning outcomes

    • By the end of this module you will have acquired the following knowledge and skills: 

      • Compute marginal and conditional distributions from joint distributions
      • Construct transformations of random variables and prove their distributions
      • Explain what is a moment generating functions and apply it appropriately
      • Prove and apply Chebyshev's inequality and the weak Law of Large Numbers
      • State and apply the Central Limit Theorem  
      • Carry out 1- and 2-sample t-tests, F-tests, permutation tests and matched pairs t-tests
      • Compute confidence intervals for unknown parameters of given distributions based on random samples
      • Carry out chi-squared goodness of fit tests for samples from specified population models
      • Carry out tests of association for contingency tables 
      • State the main properties of a bivariate normal distribution
      • Explain the relationships between distributions related to the normal distribution 
      • Use the relationships between random variables to simulate values from distributions 

  • Assessment

    • The final exam for this module counts 80%. It will be a face-to-face exam. Besides that, you will be assessed by two online quizzes in weeks 7 and 12. Each quiz counts 10%. 

      In the January exam, you will have 3 hours to complete your work. For the exam as such you probably need only 2 hours.

      The in-term assessments (quizzes) will test your knowledge of the material covered in the lectures and tutorials.  Moreover, there will also be practice online quizzes which do not count for your module result, in week 3,5, and 10, but are meant to give you some further examples of what is relevant for this module.

      To prepare for the exam, go carefully through all lecture notes, coursework solutions and the solutions of all 5 quizzes. Make sure that you carefully revise the contents of lecture notes, coursework questions and quizzes and that you can do calculations  similar to those in courseworks and quizzes on your own.

      Finally, after you did all the above preparations, you may also look at last year's exam paper as a practice test. If you need hints on the solutions of that paper, these will be given in the revision lectures. 



  • More about in-term assessment

    • There will be two assessed quizzes for this module in weeks 7 and 12.
    • Each quiz counts for 10% of your module mark.
    • Each quiz will have several multiple choice or fill-the-blank questions.
    • The additional quizzes quizzes will become live in Weeks 3, 5, and 10.
    • You will have up to 5 days to complete each quiz.
    • The examinable material in each quiz will be stated in its description.
    • Your work should be entirely your own, and must be submitted on QMplus. 
    • The quizzes will appear in the main part of the QMplus page (Module Content) under the relevant weekly session.

  • Week 1

    • Hand-written notes 1 File
      5.2 MB

    • During this week's lectures we will introduce the module, revise some basic univariate probability theory (from your first year) and introduce the conditional distribution of discrete random variables.  

       

      week 1 - no tutorial--TUTORIALS will start next week

      Since this is week 1, we will just hand out the first coursework, which you should carefully study yourself.
      Tutorials will start in week 2. In next week's tutorial we will discuss the solutions of  Coursework 1.  
      Coursework sheets are for you to practice the relevant material. You should study them but they are not assessed. 

      Reading: 

      • Make sure you are fluent in the revised (first year) probability knowledge that is required in order to follow this module. 
      • Study thoroughly the teaching material of this week on the conditional probability of events and discrete random variables.  
      • Revise your knowledge on double integration, since it will be fundamental for the introduction of joint distributions of continuous random variables next week.

      Training: 

      • Attempt all questions in Coursework 1 prior to next week's Tutorial.

    • Coursework 1 File

      Please look at this prior to the tutorial taking place next week. Try to solve it yourself.

      Solutions to be discussed in the tutorial on Friday 7 Oct.

      [In this coursework, apart from the exercises, there is useful information about important example probability distributions that we will use in this course]

      146.6 KB
    • Solutions to Coursework 1 File
      172.3 KB

  • Week 2

    • Hand-written notes 2 File
      4.6 MB
    • In this week's lectures, we will cover the joint and marginal distributions of continuous random variables.   

      In this week's tutorial, we discuss last week's Coursework 1 

       Reading: 

      • Study thoroughly this week's material on continuous random variables.
      • Being fluent in calculations of joint probabilities is crucial for understanding  more complicated marginal and conditional distributions that will be introduced next week. 
      • Familiarise yourselves with the concept of marginal distributions, as they are the building blocks of independence of random variables, which will be taught next week.

      Training: 

      • Attempt all questions in Coursework 2 prior to next week's Tutorial.

    • Coursework 2 File

      Please complete prior to Tutorial next week.

      To be discussed in Tutorial of week 3.


      78.1 KB
    • Solutions to Coursework 2 File
      124.3 KB

  • Where to get help

    There will undoubtedly be times during the term when you get stuck doing your homework or project. This is normal. 

    Come to the tutorial and we can discuss. 

    There are also sometimes problems that very much go beyond the field of maths. You can find help here: 

    QMUL Advice and Counselling Service


  • week 3

    • Hand-written notes 3 File
      4.8 MB

    • In lectures, we will continue the study of continuous random variables: covariance, marginal and conditional distributions. We will also introduce the concept of independence of random variables

      In the tutorial, we shall discuss last week's Coursework 2.

      Reading: 1. Familiarise yourselves with the concept of independence of random variables, since we will study this topic in further depth next week.

      2. Revise the definition of cumulative distribution functions; we will see how to use them to obtain transformations of random variables next week.

      Training: 

      • Attempt all questions in Coursework 3 prior to next week's Tutorial.


    • Coursework 3 File

      Please complete prior to Tutorial next week.

      To be discussed in Tutorial of week 4.


      94.5 KB
    • Solutions to Coursework 3 File
      109.2 KB

  • Week 4

    • Hand-written notes 4 File
      4.4 MB
    • In lectures, we will continue the study of independence of random variables and learn how to transform random variables via their cumulative distribution functions.

      In the tutorial, we shall discuss last week's Coursework 3.

      Reading: 

      • Become fluent in the independence of random variables, since it is crucial for the forthcoming material.
      • Revise your (first-year) probability & calculus background, as it will become much more involved next week: 
        • the definition of expectation (for continuous random variables). 
        • the rules of differentiation of one variable (product rule, chain rule, etc) and partial derivatives (Jacobian, etc) when we have more than one variable.

      Training: 

      • Attempt all questions in Coursework 4 prior to next week's Tutorial.


    • Coursework 4 File

      Please complete prior to Tutorial next week.

      To be discussed in Tutorial of week 5.


      85.4 KB
    • Solutions to Coursework 4 File
      112.4 KB

  • Week 5

    • Hand-written notes 5 File
      5.1 MB
    • In this week's lectures, you will be taught direct transformations of one or more random variables and moment generating functions. Using the latter, we will also study the sums of independent random variables.

      In the tutorial, we shall discuss last week's Coursework 4.

      Reading: 

      • Read carefully the material on transformations of random variables, as it will be fundamental for the second (statistical) part of the module. 
      • You should also become fluent in the concept of independence of random variables and the properties of their sums (expectation, variance, etc).

      Training: 

      • Attempt all questions in Coursework 5 prior to next week's Tutorial.


    • Coursework 5 File

      Please complete prior to Tutorial next week.

      To be discussed in Tutorial of week 6.


      96.3 KB
    • Solutions to Coursework 5 File
      144.5 KB
    • Week 5 Quiz

      This is a quiz testing your knowledge of the weeks 2-4 material. It does not count for your module mark. However, it will be recorded whether you took part and which result you got.

      Topics include material from weeks 2-4: Marginal and Conditional distributions of continuous random variables, Covariance and Correlation of random variables, Independence of events and random variables.

      The quiz 

      Opens: Friday, 27 October 2023, 1:00 PM

      Closes: Wednesday, 1 November 2023, 6:00 PM



    • Solution notes quiz 2 File
      233.2 KB

  • week 6

    • week 6 - lecture

      In the first lecture of this week, we shall conclude the probabilistic part of the module with some estimates of probabilities, the Law of Large Numbers and the Central Limit Theorem. 

      In the second and the third lecture, we shall discuss solutions to CWs 5 and 6 and the structure of the probability part of the exam

      In the fourth lecture of the week, we shall introduce the statistical part of the module will be introduce

      week 6 TASKS

      Reading: 

      • Read carefully the material on the Central Limit Theorem, as the normal approximation will be heavily used in the second (statistical) part of the module.
      • Lecture notes on statistics, pages 1-5.


    • Week 6 - Lecture notes - Probability part File

      These notes accompany week 6 live lectures

      149.2 KB
    • Hand-written notes 6 File

      For comments concerning the probability part of the January exam see the last page of these notes.

      3.1 MB
    • Coursework 6 File

      Please solve as many problems as you can.

      To be discussed in Tutorials of week 6.


      80.2 KB
    • Solutions to Coursework 6 File
      120.2 KB
    • Slides - Statistics part File
      429.5 KB

  • week 7 - Reading week

    •            Reading week - no lectures or tutorials. However, there will be an online test during week 7 which counts for 10% of your module result.

  • week 8

    • week 8 - lecture

      In this week's lectures we will finish covering hypothesis tests and confidence intervals for the normal distribution. We will introduce the t distribution and use it to conduct tests and construct confidence intervals for the mean when the variance is unknown. We will introduce the concept of p-values as an indication of the strength of evidence against the null hypothesis. Finally, we will consider hypothesis tests and confidence intervals for a binomial proportion and a Poisson mean using a normal approximation.

      Lecture notes Statistics part pages 5-11

      week 8 - tutorial

      We will be discussing examples 1.1, 1.2, 1.3, 1.4, 1.5 and 1.6 from the lecture notes, as well as Coursework 7.

    • Coursework 7 File


      93.4 KB
    • Solutions to Coursework 7 File
      147.1 KB

  • week 9

    • week 9 - lecture

      In this week's lectures we will be covering Goodness of Fit tests. These are tests to decide whether a sample of data can be assumed to come from a particular distribution. We will cover Goodness of Fit tests for discrete and continuous random variables.

      Lecture notes Statistics part pages 12-18.

      week 9 - tutorial

      We will be discussing Examples 2.1 and 2.2 from the lecture notes and Coursework 8.

    • Coursework 8 File
      63.8 KB
    • Solutions to Coursework 8 File
      122.9 KB

  • week 10

    • week 10 - lecture

      In this week's lectures, we will cover hypothesis tests where two samples are involved. In particular, we will cover the two-sample t-test for comparing two means and the F-test for comparing two variances. This is covered on pages 19-24 of the lecture notes of the Statistics part of the module.

      week 10 - tutorial

      We will be discussing Coursework 9.

    • Coursework 9 File
      37.8 KB
    • Solutions to Coursework 9 File
      122.6 KB
    • WEEK 10 QUIZ - Practice

      Please ensure you complete this week's quiz which serves as extra practice.

      It covers the material in Chapters 1 and 2 of the Statistics part of the notes. 

      1) Hypothesis testing and constructing confidence intervals with normally distributed random samples.

      2) Goodness of Fit tests for discrete and continuous probability distributions.

      Solutions will be released on Thursday 7 December.

    • Quiz 4 Solutions File
      129.8 KB
    • Lecture handwritten notes File
      1.0 MB

  • week 11

    • week 11 - lecture

      We will discuss contingency tables and tests about proportions and correlations. We will look at the problem of testing if two categorical variables are independent.

      week 11 - tutorial

      We will be discussing Coursework 10.

    • Coursework 10 File
      57.8 KB
    • Coursework 10 Solutions File
      125.4 KB
    • Handwritten notes File
      2.2 MB
    • Test your knowledge File
      48.9 KB

  • week 12

    • week 12 - lecture

      In this week's lectures, we will discuss contingency tables, the likelihood ratio and its relation to hypothesis testing.

      week 12 - tutorial

      We will be discussing the previous exam papers from January 2022 and January 2023.

    • p-values2 File
      42.6 KB
    • Handwritten notes File
      825.0 KB

  • Assessment information

    • The final exam for this module counts 80%. It will be a face-to-face exam. Besides that, you will be assessed by two online quizzes in weeks 7 and 12. Each quiz counts 10%. 

      In the January exam, you will have 3 hours to complete your work. For the exam as such you probably need only 2 hours.

      The in-term assessments (quizzes) will test your knowledge of the material covered in the lectures and tutorials.  Moreover, there will also be practice online quizzes which do not count for your module result, in week 3,5, and 10, but are meant to give you some further examples of what is relevant for this module.

      To prepare for the exam, go carefully through all lecture notes, coursework solutions and the solutions of all 5 quizzes. Make sure that you carefully revise the contents of lecture notes, coursework questions and quizzes and that you can do calculations  similar to those in courseworks and quizzes on your own.

      Finally, after you did all the above preparations, you may also look at last year's exam paper as a practice test. If you need hints on the solutions of that paper, these will be given in the revision lectures. 

      FEEDBACK: Solutions to a quiz is provided shortly after the end of the quiz. You are also more than welcome to ask questions concerned with the problems in quizzes and their solutions - just email us and we shall try to answer you questions as soon as we can.

  • Past papers

  • Teaching arrangements

    • There are four timetabled  sessions each week: 3 lectures and 1 tutorial.
      The two (1-hour) live sessions on Thursdays 11:00-13:00 and one on Fridays 13:00-14:00 will be lectures, introducing the main ideas and fundamental knowledge of the relevant topics along with some examples. 
      The second hour on Friday 14:00 - 15:00 will be a tutorial. (Week 1 is exceptional since in this  week a lecture will be used for and additional lecture).

      All lectures and tutorials will be in person in the Great Hall.  If you are not able to attend, you will be able to watch a recorded version on Q-Review. 

  • Hints and tips

    • To be successful in this module

      • Read the assigned reading thoroughly. Make notes. 
      • Review lecture notes and, if necessary, use one of the recommended survey books to enhance your understanding of the week’s topic.
      • Always attempt the coursework questions each week.  
      • Bring your solutions attempts into the tutorial with you.

      Tip: Make it a goal to understand each learning item during the week in which it was taught.


  • Coursework

    • You should always come prepared to tutorials, have attempted all questions in the coursework, so that you can ask your questions and pay attention to the parts where you struggled.  The tutorials form a core learning experience of the module. You should study the coursework questions carefully and work on them but you do not need to hand in solutions. These courseworks will not be assessed. Information about assessment can be found in the next section.

  • Q-Review

  • Reading List Online