Topics
- Bayes Theorem and the Monty Hall problem - adored this part
- Probability: sample space, events, probability measure, partitioning the sample space
- Binomial experiments
- Random variables, expected value, variance, standard deviation
- Continuous vs discrete random variables, probability density function
- Poisson, exponential, uniform and Gaussian distributions
- Joint probability distributions
- Sample mean, sample variance, law of large numbers, central limit theorem, confidence intervals (except we didn't really do anything about hypothesis testing, confidence intervals were just mentioned in the slides. If only we'd done stuff about C.I.s and hypothesis testing!)
- Chi-square: we didn't do what a lot of statistics classes do, which is get data and do the chi-square test on it. Instead, we'd be given some data and a proposed model e.g. y = ax^2, and have to work out from the definition of chi square what the parameter a was which minimised chi square and then use the parameter to calculate chi square and see how good the model is. It's kinda cool being able to do that, to just go from the general shape of the model to find its exact parameter(s) and test how good they are, but a) it was unfortunate that barely anyone else seems to do this harder version and so I couldn't really find any resources online to help b) it is very very finicky, with so many squares and dividing by tiny numbers that a small arithmetic error can result in an answer 1000x off the actual answer.
- A little bit of algorithms, contrasting Quicksort and Bubblesort.
- Markov chains - a Markov chain is one in which the probability of being in state i given that you were in state j is equal to the probability of being in state i given that you were in state j and before that in state k and before that in state .... so basically only the previous state matters. We learned how to construct a Markov matrix given some of these probabilities, how to get the probability of a certain outcome several steps on, how to find the longterm steady state (eigenvector) of the system, the detailed balance conditions, how to construct a matrix given its eigenvector, and absorbing states. I liked these problems once I figured out how to deal with eigenvectors - it's quite fun.
Review
Very frustratingly, the lectures, homework and exams seemed to be based off three different curricula. Fortunately the exams were the easiest of the three but it was very stressful having half the homework questions just be random impossible things we hadn't been taught. The lecturer would be doing something on the board and it'd get too long or complicated and so he'd say 'I'll just ask it in the homework'. Wat. That's not the point of homework. The homework was often extremely difficult in parts and often had components that weren't taught until the day before the homework was due, so I couldn't do it in advance as I would've liked, and had to just bang my head off the wall until eventually in the tutorial me, people around me and the TA came up with something together. Such a contrast from the lovely Multivariable Calculus homeworks, in which we didn't have to guess at what the question could possibly mean.
The exam tended to have Chi Square and Markov matrix come up every year, often with a Bayes rule diagnostics question and probability theory question. So the exam was a lot more practical, easier, and more in line with what the course booklet said.
The lectures, on the other hand, were super difficult and confusing, often proofs-based (which is something maths students, not science students, are there for), and seemed super aimless. The lecturer would often start a topic only to move onto another one in the next class and then come back to the first one, so the arrangement of my notes made little sense and I eventually stopped rewriting my notes after class (I'd write them during the class but wouldn't rewrite them afterwards as I would for other classes) because it was just so frustrating. Another big problem was that the lecturer, much as he seemed like a nice dude, couldn't answer our questions and the class was really struggling. Close to exams he said not to worry because people don't fail his exams so I really have to wonder what the marking scheme is like because a lot of us didn't have the slightest clue. In fairness I did get a good handle on the things that came up, i.e. probability theory, Bayes rule (<3), Chi square and Markov chains (<3), but I was very glad the papers were so predictable because if he'd asked the pretty structureless stuff he talked about in his lectures it would've been very very stressful and I don't know if I could've done well.
Final result: 92%; 93% in exam.
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