Overview
With the laws of large numbers and the central limit theorems, what people mix up is not the conclusions but the conditions: which one needs i.i.d.? Which only needs pairwise uncorrelated? Which doesn’t even need a variance?
This part first sets up the tool — the characteristic function — then lines up the conditions of the limit theorems side by side in tables, and finally walks through the characteristic-function proofs of Khinchin’s LLN and the Lindeberg–Lévy CLT.
Overview
The chapter on random vectors is where my notes have the densest cluster of “know the proof” marks: Chebyshev’s inequality, the Schwarz inequality, the necessary and sufficient condition for a correlation of ±1, non-negative definiteness of the covariance matrix, the law of total expectation…
The good news is that these proofs share one idea: build a non-negative quantity, then ask when it equals 0. Keep an eye out for that thread as you read.
Overview
The common distributions are the foundation of all of probability: characteristic functions, sampling distributions and parameter estimation later on all lean on their means, variances and relationships.
The goal for this part is “derive it + remember it”. I suggest starting with the relationship map in Section 5: once you remember identities like \(Ga(1,\lambda)=Exp(\lambda)\), \(Ga(\frac n2,\frac12)=\chi^2(n)\) and \(Be(1,1)=U(0,1)\), many distributions can be derived from one another instead of memorized. For each distribution you should be able to write down the pmf or density, the mean and the variance; ☆ marks the key ones.
About this series. These are my review notes on Probability Theory and Mathematical Statistics by Mao Shisong, Cheng Yiming and Pu Xiaolong (茆诗松《概率论与数理统计教程》, “the Mao book” below), a standard Chinese textbook for graduate entrance exams and PhD qualifiers. There are 7 parts.
P××× = page in the textbook; “Guide” = its companion study guide and solutions manual.
How to use this part. Short-answer questions often ask you to “state the definition of …”. Each concept below gives an exam-ready definition you can write down directly, plus a one-line intuition.
At the very begining, congradulationgs to this homepage!
Some course notes will comming soon, welcome to contact me at any time, looking forward to chatting with you!
