Advanced Probability and Statistics
MAFS 5020 (L1)
Summer, HKUST
Instructor, TA and accesses for help:
Instructor: Kani Chen.
All students are strongly encouraged to get help from the instructor (or the TA). In
particular, they are encouraged to ask questions in class and talk to the instructor
right after each class. Other accesses for help are:
email to the instructor at “makchen@ust.hk”, or drop by the instructor’s office at
Room 3426 any time, or call the instructor at 2358-7425;
M ain Ref erences:
1. Lecture notes. (All exams problems will be based on the contents covered in lectures.)
2. “A Course in Probability Theory”, 3rd edition, by Kai Lai Chung; Publisher: Academic press. ISBN: 0121741516.
3. “Probability: Theory and Examples”, 3rd edition, by Richard Durrett; Publisher:
Thomson Brooks/Cole. ISBN: 0534424414.
4. “Mathematical Statistics”, 2nd edition, by Jun Shao; Publisher: Springer. ISBN:
0387953825.
5. “Mathematical Statistics: Basic Ideas and Selected Topics.” by Peter Bickel and Kjell
Doksum; Publisher: Prentice Hall. ISBN: 0135641470.
Gradings and Exams:
Midterm 50%, Final 50%.
Course Contents:
PART 1. Probability Theory:
1. Probability calculation; probability space; set algebra; conditional probability;
independence; Bayesian formula; indicator functions.
2. Random variables; distribution and density; expected value; independence; conditional distribution and density; conditional expectation; moments; variance and
covariance; Typical discrete and continuous distributions.
3. Weak and strong convergence of random variables; convergence in distribution;
Borel-Cantelli lemma; Fatou’s Lemma; dominated convergence and monotone
convergence; Jensen’s inequality; Markov, Chebyshev and Kolmogorov inequalities.
4. Weak and strong laws of large numbers.
5. The central limit theorem; characteristic functions.
6. Martingale: definitions and main results.
PART 2. STATISTICS:
1. Formulation of statistical models.
2. Parametric models; Point estimation; method of moments; maximum likelihood
estimation; unbiaseness; UMVUE; Cramer-Rao lower bound; sufficiency and completeness.
3. Efficiency, Fisher information.
4. Interval estimation.
5. Test of hypothesis; Neyman-Pearson lemma.
6. Linear models.
7. Nonparametric models.
Remark: The course contents may be subject to changes depending on the progress
of teaching.