Problems for Thursday, January 25th: Note that with the exception of 1.1 and 1.2, these problems assume that you know Theorem 1.1. Also, it's perfectly acceptable (and even recommended!) to use software to calculate powers of matrices.
Page 78, exercises: 1.1, 1.2, 1.4, 1.5, 1.6, 1.7.
Problems for Wednesday, January 31st (note new date): 1.8, 1.9a)b) (there's a misprint in c), 1.16, 1.18, 1.26, 1.29. When you are asked about long time (limiting) behavior, it's OK to rely on computer experiments (we don't quite have the machinery yet to deal with these things theoretically).
Problems for Wednesday, February 7th:
From the textbook: 1.10, 1.11, 1.13, 1.38. 1.39.
Mandatory assignment 2023, problem 1.
Problems for Wednesday, February 14th:
From the textbook: 1.14, 1.40, 1.42, 1.59, 1.61, 1.67.
Exam 2023, Problem 1. In question d), "reversible" means that the time reversed process has the same transition probabilities as the original.
Problems for Wednesday, February 21st:
From the textbook: 1.44, 1.48, 1.50 (opprinnelig utelatt ved en trykkfeil), 1.51
Exam 2004, problem 1. (Note that this problem uses a different convention for the transitions probabilities than our textbook: In this problem, \(p_{ji}\)is what we call \(p_{ij}\).)
Problems for Wednesday, February 28th:
From the textbook: 1.52, 1.53, 1.56, 1.67, 1.70
Mandatory assignment 2023, problem 2.
Problems for Wednesday, March 6th:
From the textbook: 1.72, 1.73, 2.1, 2.4, 2.5, 2.9
Exam 2020, problem 3 (most of this problem is theory you already know).
Problems for Wednesday, March 13th:
From the textbook: 2.15, 2.16, 2.19, 2.21, 2.22, 2.24, 2.26
Problems for Wednesday, April 3rd (there are no classes in the midterm week March 18–22):
From the textbook: 2.27, 2.28, 2.30, 2.34, 2.39, 3.2, 3.4, 3.17, 3.17
Problems for Wednesday, April 10th: Note. I now see that I was a little ahead of myself when I gave these problems. They involve stationary distributions which we will discuss in the lecture later on April 10. However, the only thing you need to know about stationary distributions \(\pi\), are that they are solutions to the (vector) equation \(\pi Q=0\). It's also useful to know that according to Theorem 4.8, distributions always converge to the stationary distribution (granted that the chain is irreducible.)
From the textbook: 4.1, 4.2, 4.5, 4.8, 4.14
Wednesday, April 17th: Old exam problems this time. Be aware that notation has changed over the years, and that the matrix we call \(Q\) (the infinitesimal generator) is often referred to as \(\mathbf{R}\) in older problems. It also seems that earlier years students have had less information about convergence to stationary distributions than we have. For us, it usually suffices to appeal to Theorem 4.8.
Wednesday, April 24th:
From the textbook: 4.16, 4.17, 4.18, 4.19, 4.22, 4.23, 4.26.
Wednesday, May 8th (there are no classes on May 1st):
From the textbook: 4.27, 4.28, 4.30, 4.32
From the note on Brownian motion: Section 1, problems 1, 2.
Wednesday, May 15th:
From the note on Brownian motion: All problems from sections 2, 3, and 4.
Exam 2016, Problem 3 (in question b, "standard Brownian motion" is just Brownian motion as defined in note on Brownian motion).
Wednesday, May 22nd:
Exam August 17th, 2023 ("konteeksamen"). You may want to treat this as a trial exam. There is a set of solutions here, but don't look too early.