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: Dr Gordon Slade
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.
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