Monday, August 19. Started with a mixture of sections 1.1 and 1.2. Introduced\(\sigma\)-algebras and probability measures and went through the results in Remark 1.2 and Section 1.2. Next time I shall return to Section 1.1 and the heavier results around the Monotone Class Theorem. Podcast. Notes from the lecture (in Norwegian).
Thursday, August 22. Continued lecturing on a mixture of sections 1.1 and 1.2. The only thing left now is the proof of the Monotone Class Theorem which I'll do on Monday. I'll not cover section 1.3 in class (it consists mainly of examples, some of which I'll done already), but will continue with sections 1.4 and 1.5. Podcast. Notes
Monday, August 26. I first finished the first two sections by proving the Monotone Class Theorem and continued by introducing conditional probabilities and proving Bayes Theorem. Finally, I introduced the (important!) notion of independence and showed by example that a pairwise independent set of events need not be independent. On Thursday, I'll start Chapter 2. Podcast, part 1. Podcast, part 2. Notes
Thursday, August 29. Before the break, I went through Section 2.1 up to (and including) the definition of distribution functions, filling in some details along the way. After the break, I did a selection of this week's exercises: 1.5, 1.6, 1.8, 1.9, 1.16, and 1.18. Podcast. Notes.
Monday, September 2. The physical lecture was canceled, but I covered the material through podcacts. The first podcast deals with the results on page 41 and 42. The second covers the rest of Section 2.1 plus section 2.2 and 2.3.
Podcast, part 1, Notes, part 1.
Podcast, part 2 Notes, part 2.
Thursday, September 5. Went quickly through Section 2.4 and started Section 2.5 where I got to the bottom of page 55. After the break, I did a selection of this week's problems: 1.31, 1.34, 1.42, 1.44 and the beginning of 2.2. As there seemed to be interest in a full solution of exercise 2.2, I have made one available here. Podcast. Notes.
Monday, September 9. I started by proving Theorem 2.28. Then I went straight to section 3.1 to cover the expectation of a general stochastic variable while we still had Theorem 2.28 fresh in mind. I gave the definition of the expectation in the general case and proved theorem 3.4. Next time, I will first pick up a few bits and pieces from sections 2.5 and 2.6 before I continue with the (very important) Theorem 3.5. Podcast. Notes.
Thursday, September 12. Went back to section 2.6 to pick up the definitions of moments, variance, and standard deviation, before I continued with Theorem 3.5 on the expectation of products of random variables and its corollary 3.6. After the break I did problems 2.11, 2.16, 2.17, and 2.20. On Monday I shall continue with Section 3.2 (which I'll treat in a rather cursory fashion) and sections 3.3 and (the beginning of) 3.4. Notes. Podcast lecture. Podcast problems.
Monday, September 16. A most productive day! I first went quickly through Section 3.2, leaving the proof of Theorem 3.9 as self-study, and then continued with sections 3.3 and 3.4, following the book quite closely. This means that I have caught up with the schedule and will start Chapter 4 next time. Notes. Podcast.
Thursday, September 19. Lectured on Section 4.1 up to and including Proposition 4.6. After the break I did exercises 3.3, 3.5, 3.4, and one half of 3.6 (in that order). If you thought they were tough, so did I! Notes. Podcast
Monday, September 23. I started the lecture with a little detour to sort out the relationship between \(\sum_{n=1}^{\infty}P(A_n)=\infty\) and \(\prod_{n=m}^{\infty}(1-P(A_n))=0\) that is needed in the proof of the second half of Borel-Cantelli's lemma. I then followed the book rather closely from Definition 4.8 up to and including Lemma 4.11. Notes. Podcast.
Thursday, September 26: Before the break, I lectured on the Monotone Convergence Theorem and Fatou's Lemma (and the corollary in-between). After the break, I did problems 2.40, 2.41, 2.43, and 2.50. Notes. Podcast. (For some reason, the microphone fell out for about a minute around 1.03).
Monday, September 30. Finished Chapter 4 by proving Theorem 4.18 (Dominated Convergence) and its corollaries and sketching the argument in Section 4.3. After the break I talked a little bit about different laws of large numbers and proved theorems 5.1 and 5.2. Next time I shall continue with section 5.4. Notes. Podcast of first half. Podcast from second half.
Thursday, October 3. Before the break, I covered section 5.4. After the break I did exercises, concentrating on the convergence in probability part of exercise 4.4, which is technically quite challenging. Notes. Podcast.