# 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 420

Real Analysis I

“F just comes along for the ride…”

Text: Real Analysis: Modern Techniques and Their Applications by Gerald B. Folland

Prof: Dr Stephen Gustafson

Prof Gustafson is an engaging professor. He is able to give a quick, intuitive overview of quite complex topics in class, leaving us to work out the technical details on our own. He also is very helpful during office hours. He also posted his notes online.

Difficulty

The classes move very fast so one might find it challenging keeping up with the readings. Also, though the classes are easy to follow, the homework is increasingly challenging and time-consuming. The final exam was quite hard, practice and recalling theorems and definitions are all important. This course covers a lot of material, so watch out!

Key Concepts

Sigma-Algebras

Measures

Differentiation of Measures

Lebesgue integration

Convergence of Functions

LP spaces

Hard Concepts

Lebesgue Radon Nikodyn Theorem: Quite powerful theorem, need to notice where to apply it. Also can be hard to find the decomposition at times.

Fatou’s Lemma: Tricky working with the limit infimum, also quite powerful and pops up in unexpected places.

Conclusion

One of the harder courses I have ever taken. Started learning some really powerful theorems about convergence, for example.