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MATH 408 Mathematical Statistics






Syllabus [pdf]

MWF 12:00-12:50 pm in ZHS 352


KAP 470A
Email (please include “408” in the subject line)
Office Hours MWF 1:30-2:30 pm, or by appointment
Discussions TTh 9:00-9:50 am, 10:00-10:50 am in KAP 165
Teaching Assistant Grigory Sokolov (, KAP 415)

Coure Description
Mathematical Statistics is the branch of applied mathematics that studies ways of drawing inferences from limited data. The main goal of this course is to give an introduction to the fundamental concepts, ideas, and methods of Statistics.

MATH 407 Probability Theory

J. Rice, Mathematical Statistics and Data analysis, 3rd edition, 2007.

Course Plan
The plan is to cover most of Chapters 6-11. Topics will include: the sample mean and the sample variance and their properties; estimation of population parameters under simple random sampling and stratified random sampling; confidence intervals; fundamental concepts of statistical inference; method of moments; method of maximum likelihood; Cramer-Rao lower bound; sufficient statistics; testing hypothesis; likelihood ratio test and goodness of fit; probability plots; descriptive methods for summarizing data; comparing two samples; parametric and non-parametric methods.





First Midterm
Second Midterm


Homework problems and due dates will be posted here. These problems will be collected in Discussion class on Thursday. Late homework will not be accepted for any reason.

A quiz will be given every other Tuesday (for exact dates see “Important Dates” below). All quizzes are closed-book. Calculators are allowed.

Midterm Exams
There will be two (one hour) midterm exams: Wednesday, February 13 (exam 1) and Wednesday, March 27 (exam 2). The 2nd exam will cover the material after the 1st exam. Both exams will be given in regular class time. The place will be announced later. Both exams are closed-book, but you are allowed to bring one sheet of formulas. Calculators are allowed.

Final Exam
The final exam will be comprehensive and it will be held at the time specified in the University Schedule of Classes: Friday, May 10, 11am-1pm, location to be announced. The final exam is closed-book, but you are allowed to bring one sheet of formulas. Calculators are allowed.

Important Dates
Jan 22, Feb 5, 19, Mar 5, Apr 2, 16, 30
Jan 25, Feb 8, 22, Mar 8, 29, Apr 12, 26
First Midterm
Wednesday, February 13
Second Midterm
Wednesday, March 27
Friday, May 10, 11am-1pm

Official announcements, homework assignments, due dates, solutions will be posted on the course website. You are expected to check the course website on a regular basis. You are encouraged to read the appropriate sections of the textbook in advance and discuss the homework assignments with other students.

Behavior that persistently or grossly interferes with classroom activities is considered disruptive behavior and may be subject to disciplinary action. Such behavior inhibits other students’ ability to learn and an instructor’s ability to teach. A student responsible for disruptive behavior may be required to leave class pending discussion and resolution of the problem and may be reported to the Office of Student Judicial Affairs for disciplinary action. In particular, the use of cell phones during class or conversation is disruptive behavior.

Academic Integrity
All students are responsible for maintaining standards of academic integrity. The university regards cheating as a very serious issue and recommends F in the course for any violation. In particular, collaboration, use of notes, or any electronic devices during quizzes, midterms or the final are strictly prohibited.


All slides in a single [pdf]

Lecture 1: ABC of Probability [pdf]
Lecture 2: Conditional Probability [pdf]
Lecture 3: Discrete Random Variables [pdf]
Lecture 4: Continuous Random Variables and Transformations of Random Variables [pdf]
Lecture 5: Joint Distributions [pdf]
Lecture 6: Expectation, Variance, Covariance, and Correlation [pdf]
Lecture 7: Conditional Expectation and Conditional Variance [pdf]
Lecture 8: Inequalities [pdf]
Lecture 9-10: The Law of Large Numbers & The Central Limit Theorem [pdf]
Lecture 11: Probability Theory: an Overveiw [pdf]
Lecture 12: Introduction to Survey Sampling [pdf]
Lecture 13-14: The Sample Mean and the Sample Variance Under Assumption of Normality [pdf]
Lecture 15: Accuracy of estimation of the population mean [pdf]
Lecture 16: Estimation of the Population Variance [pdf]
Lecture 17: Normal Approximation to the Sampling Distribution of the Sample Mean [pdf]
Lecture 18: Estimation of a Ratio and the Delta Method [pdf]
Lecture 19: Stratified Random Sampling [pdf]
Lecture 20-21: Neyman Allocation vs Proportional Allocation and Stratified Sampling vs Simple Sampling [pdf]
Lecture 22: Survey Sampling: an Overview [pdf]
Lecture 23: Fundamental Concepts of Modern Statistical Inference [pdf]; The Method of Moments [pdf]
Lecture 24-25: The Method of Maximum Likelihhod [pdf]
Lecture 26-27: Confidence Intervals from MLEs [pdf]; The Bootstrap Method: Simulation Results [pdf]
Lecture 28: Efficiency and the Cramer-Rao Lower Bound [pdf]
Lecture 29-30: Testing Hypotheses: The Neyman-Pearson Paradigm [pdf]
Lecture 31: Generalized Likelihood Ratio Tests [pdf]
Lecture 32-33: Pearson's Chi-Squared Test for Multinomial Data [pdf]

Lecture 34: Summarizing Data [pdf]
Lecture 35: Summarizing Data II [pdf]
Lecture 36: Summarizing Data III [pdf]
Lecture 37: Gapminder
Lecture 38: Fundamental Concepts of Statistical Inference [pdf]


Late homework will not be accepted for any reason.
(Learn to do things in advance)

Due Date
Conditional Probability

Ch. 1, # 7, 45, 65, 75, 77.

Jan 31
Random Variables, Joint Distributions, and Expectations Ch. 2, # 31, 59;
Ch. 3, # 15;
Ch. 4, # 21, 49, 67.
Feb 14
Limit Theorems and Distributions Derived from the Normal Distribution Ch. 5, # 13, 15, 19 (for #19, provide a source code you used)
Ch. 6, # 3.
Feb 28
Survey Sampling Ch. 7, # 1, 18, 21.
March 14
The Method of Moments & The Method of Maximum Likelihood Ch. 8, # 5(a,b,c), 7(a,b).
April 4
Testing Hypotheses Ch. 9, # 1, 7, 9, 17(a,b)
April 18
Summarizing Data Ch. 10, # 1, 11, 16, 19
May 2


Homework 1 [pdf], mean=9.55
Homework 2 [pdf], mean=7.53
Homework 3 [pdf], mean=6.11
Homework 4 [pdf], mean=8.45
Homework 5 [pdf], mean=9.10
Homework 6 [pdf]

Quiz 1 [pdf], mean=8.81/10
Quiz 2 [pdf], mean=6.72/10
Quiz 3 [pdf], mean=8.26/10
Quiz 4 [pdf], mean=8.28/10
Quiz 5 [pdf], mean=5.04/10
Quiz 6 [pdf], mean=6.25/10

Midterm 1 [pdf], mean=75.72/100, std=15.95
Midterm 2 [pdf], mean=78.45/100, std=17.32


  • "The quiet statisticians have changed our world, not by discovering new facts or technical developments but by changing the ways we reason, experiment, and form our opinions," Ian Hackling. Well, statisticians are not quiet any more! Watch The Joy of Stats, a very inspiring video with Prof. Hans Rosling.
  • Hans Rosling shows the best stats you have ever seen. Download the gapminder, and play with it. It is fun!
  • If you want to quickly refresh your knowledge of Probability and Statistics, these notes by Prof. Cosma Shalizi might be useful. These notes are short and contain the essence.
  • If you like Statistics, I would strongly recommend you to read a non-technical book The Cartoon Guide to Statistics by Larry Gonick and Woollcott Smith. Here is a brief and subjective review.
  • If the material we cover in class is not enough for you, and you want more, I would recommend to try All of Statistics: A Concise Course in Statistical Inference by Larry Wasserman. It is available online here. This is a very good book!