• Announcements: Today is the last class with new material this semester. As I mentioned in my email to you, the Final Exam will cover the material in Lectures 1-22. Please remember that the next (and last!) 2 lectures will be an in-person review, in preparation for the Final Exam, focusing on solving problems in the list of review problems. Please join us, and come ready with questions to ask!

• Video 1: Simulations: Download file or watch below

• Video 2: Simulating a random variable with prescribed PDF (using inverse CDF): Download file or watch below

• Video 3: Example of simulating an exponential random variable using inverse CDF: Download file or watch below

• Video 4: Simulating a random variable with prescribed PDF (using the rejection method): Download file or watch below

• Video 5: Example of simulating a standard normal random variable using rejection method: Download file or watch below

Final note: there are much more sophisticated rejection methods for simulations, which use another mathematical structure we discussed (Markov chains), and are called Markov chain Monte Carlo methods (MCMC). In short, the idea is to craft a Markov chain whose stationary distribution is the one we want to simulate. For example, the Metropolis--Hastings algorithm does this, and was invented in order to solve complicated mathematical problems crucial in the development of the hydrogen bomb.