Sessions:
16 February 2026: organisational details, revision of discrete probability from the Discrete Mathematics course
- Problems: week 1
- Solved in class: 1, 2, 3, 4, 6, 7
- Assigned as homework: 11
23 February 2026: discrete probability, conditional probability, independence
- Problems: week 2
- Solved in class: 1, 2, 3, 4, 5, 6, 11
- Assigned as homework: 8
2 March 2026: conditional probability, Bayes' formula, Simpson's paradox
- Problems: week 3
- Solved in class: 1, 2, 3, 4, 5, 8
- Assigned as homework: 10
9 March 2026: Discrete random variables, advanced example of Bayes' formula
- Problems: week 4
- Solved in class: 1, 2, 3, 4, 5, 6, 9
- Assigned as homework: 11
16 March 2026: Expectation of a discrete random variable, linearity of expectation, two envelopes problem
- Problems: week 5
- Solved in class: 1, 2, 3, 4, 5, 10, 13
- Assigned as homework: 14
23 March 2026: Variance and standard deviations, joint distributions and marginal distributions, covariance
- Problems: week 6
- Solved in class: 1, 2, 4, 6, 8, 10, 11
- Assigned as homework: 9
30 March: Revision for test 1
- Problems: week 7
- Solved in class: 1, 2 (independent work from there)
- Assigned as homework: 16
6 April 2026: no tutorial (Easter Monday)
13 April 2026: TEST 1
20 April 2026: discussion of test 1, continuous distributions and random variables
- Problems: week 10
- Solved in class: 1, 8, 10
- Assigned as homework: 6, 15
Some further notes:
- Due to test 1 and Easter, this sheet contains material from four weeks of lectures, the entirety of the module on continuous distributions. Since next week we will likely be starting with the statistics module, there is little time to dedicate to continuous distributions in the tutorials. This is not to say that they will not be prominent in test 2 or the exam, so it is in your interests to solve as much of this sheet as possible. This is also the reason why we have two homework problems this week.
- One skill crucial to this module is integration, just as finite and infinite sums were for the discrete distributions. Beyond the numbered problems, one way to practice it is with the table of named distributions before problem 5. Each of the fields that is not marked with an asterisk can be treated as an exercise: for the pdf column, integrate the expression over the real numbers and check that it evaluates to 1; for the cdf column, integrate the pdf (with the integrand substituted for x) from negative infinity to x and verify that you get the cdf; for the expectation column, multiply the pdf by x and integrate over the real numbers to check that you get the stated value; for the variance column, determine the expected value of the square of x and use that and the expectation to check that you get the stated value. The blanks for the Cauchy distribution mean that you can check that these integrals diverge, the asterisks mean that the expression given is either by definition or too difficult to verify as a standard integration exercise (the (*) for the pdf of the normal distribution means that while this is technically doable with techniques known to you, it involves some really clever trickery which is unlikely to be helpful anywhere else, though if you are curious, it is well worth looking up). One can remark, however, that for the cdf of the normal distribution you can check that the substitution z = (x-mu)/sigma leads to the integral corresponding to a N(0,1) variable as an exercise in substitution.
27 April 2026: bias and variance of estimators, use of Markov, Chebyshev and the central limit theorem to estimate probabilities, convolution
- Problems: week 11
- Solved in class: 2, 4, 8
- Assigned as homework: 3, 6
4 May 2026:
11 May 2026:
18 May 2026: TEST 2