#### 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.