# Course Review: MATH 418

#### Probability

“I heard you had a lot of difficulty with the last homework. So you will be relieved to note that next homework you will have another opportunity to practice similar problems.”

Text:  A First Look at Rigorous Probability Theory, 2nd ed by J.S. Rosenthal

Prof Slade is very clear and keeps things simple. He also has no problem slowing down to answer questions from students if they are not following along. He has a dry sense of humor, that keeps the class interesting, and he can even by funny when he is talking in earnest.

Difficulty

After taking measure theory, several sections of the course can feel like review. This was a good thing for me, as I found MATH 420 a tad fast. There are a handful of new techniques that you learn in the homework and class, but there are not that many new concepts if you have some background in measure theory and probability. MATH 421 material also comes up in terms of weak convergence.

Key Concepts

Probability Triples

Random Variables

Distributions

Expectation

Borel-Cantelli

Modes of Convergence

Law of Large Numbers

Central Limit Theorem

Characteristic Functions

Hard Concepts

Tail Events: Kind of funny to think about. Also, include definitions of limit supremum and limit infimum for sequences of event which can be difficult to convert to statements about limits of random variables.

Weak Convergence: There are a lot of equivalent statements, and if you pick the wrong one it can be a mission to prove that convergence occurs.

Conclusion

Good review of measure theory, and gives you a mathematical foundation to elementary probability.

# Course Review: MATH 302

Introduction to Probability

“Donald claims that he won the popular vote if you subtract the 3 million illegal voters. Assuming that 3 million people did vote illegally, compute the probability that Donald is correct.”

Text: Introduction to Probability by David F. Anderson, Timo Seppalainen, and Benedek Valko

Prof:  Dr. Martin Lohmann

Dr. Lohmann’s lectures largely consisted of (usually) interesting examples. Some students found his accent and his handwriting a bit challenging to follow. This, combined with the fact that he defers posting of lecture notes, made the course harder than necessary for such students. I did not have any difficulty understanding what was being said, and the few times I found his handwriting became hard to read, he clarified immediately. Also, if you ask a stupid question, expect some deft sarcasm in response!

Difficulty

While I find probability counter-intuitive, the assignments were all quite doable. Occasionally, harder questions marked with a star were provided. The first midterm was quite tricky, however, we were given a practice midterm beforehand that was conceptually quite similar to the actual exam. The second midterm and final were both significantly easier and we also were given similar practice material.

Key Concepts

Counting

Sets

Discrete vs Continuous Probability Distributions

Mean, Variance and Covariance

Joint Distribution

Convergence in Probability/Distribution

Conditional Probability

Moment-generating functions

Hard Concepts

Counting: I always make incorrect assumptions w/regard to counting. I think the key to easier problems is to identify whether you are using replacement/no replacement and order/no order. For harder problems, it is often necessary to construct a bijection of sorts or use a symmetry argument.

Conclusion

Good to get some practice with counting and probability. About as much theory as you would expect from such a class.