# STAT2911 Probability and Statistical Models (Advanced)

## General Information

This page contains information on the Intermediate Unit of Study STAT2911 Probability and Statistical Models (Advanced).

This unit is offered in Semester 1.

Lecturer(s): Uri Keich

For further information on Intermediate Mathematics and Statistics, refer to the Intermediate Handbook. In particular, see the STAT2911 handbook entry for further information relating to STAT2911.

You may also view the description of STAT2911 in the central units of study database.

- Credit point value: 6CP.
- Classes per week: Three lectures, one tutorial and one computer laboratory session.

Email enquiries about STAT2911 may be sent to STAT2911@sydney.edu.au.

Students: Please give your name and SID when emailing us. Anonymous emails will not be replied to.

Students have the right to appeal any academic decision made by the School or Faculty. For further information, see the Science Faculty web site.

## Online Resources

Detailed course information is available on the LMS site, and also here:

#### Staff

- Lectures and Tutorials: Uri Keich
- Computer Lab: Kristen Emery

#### Schedule

- Lectures: Mon 11am (Physics Lecture Theatre 1, Rm 405), Tue 11am (Chemistry Lecture Theatre 2), Wed 1pm (Carslaw 375)
- Tutorials: Mon 3pm (Mechanical Engineering Tutorial Room 2) OR Tue 1pm (Carslaw 451) OR We d 2pm (Eastern Av 310)
- Computer Lab: Tue 3pm (Carslaw 610) OR Wed 3pm (Carslaw 610) OR Wed 4pm (Carslaw 610)
- Office Hour (Carslaw 821): Mon 5-6pm

#### Grading Structure and Schedule

- Two quizzes, one in week 6 (16/4/18) and one in week 11 (10%)
- Two assignments due by (5%): 20/4/18 5pm (week 6), 1/6/18 5pm (week 12)
- Computer work handed in weekly (5%)
- A one-hour computer exam (open book) held in the last week (10%)
- A two hour final examination (70%)
- A solved problem set is due at the start of each tutorial. This work will be partially graded and these grades will be a major component in th e evaluation of any special consideration request.

#### Textbook

- Rice J.A.
*Mathematical Statisics and Data Analysis*3rd Edition.

#### References

- Feller, W.
*An Introduction to Probability Theory and its Applications*Third edition, Volume 1. - Information for both the the R computing system
- There are many user-contributed manuals/guides for R on the R website

#### Overview and Objectives

This course introduces the student to the abstract mathematical theory of probability -- a mathematical branch that was developed to model expe riments whose outcome is random. Understanding this theory is necessary for anyone who wishes to do more than follow recipes in a statistical manu al.

Concentrating on a mathematical theory means that mathematical rigor is a main theme of this course (and what largely sets it apart from STAT20 11): we progress through defining basic objects of interest, stating theorems about these objects and rigorously proving the them. As we develop t he theory we demonstrate its utility by showing what we can learn from it when analyzing the random experiments it presumably models. At the same time having an intuitive understanding of how random experiments behave helps to guide us as we develop the theory.

The course weaves statistical concepts such as estimation and construction of confidence intervals into the mathematical discussion exposing th eir theoretical foundations. The goal is to allow the students acquire a deep understanding of these notions that will allow them the basic manipu lation and modification required when analyzing previously unseen variants of well studied problems.

The course makes heavy use of R but more as a tool to help visualize and internalize the theory than as a traditional data analysis tool. As su ch there is heavy emphasis on simulation studies through which the student gains competency in using R and at the same time a better intuition of the abstract notions.

#### Learning Outcomes

Students who successfully complete this unit should be able to demonstrate that they have internalized knowledge of the following subjects:

- Probability Axioms; independence; conditional probability;
- Discrete distributions and random variables;
- Expectation; functions of rv; variance; Chebychev's inequality;< /li>
- Sums of independent discrete rv; Bernoulli variables; Binomial distribution;
- Geometric and negative binomial distributions; Poisson distribution;
- Weak law of large numbers; Poisson limit of binomial;
- hypergeometric distribution; multivariate distributions; multinomial distribution; simulat ion of discrete rv;
- Estimation of parameters using Method of Moments;
- Estimation of parameters using Maximum Likelihood;
- Unbiased estimator; Mean Square Error; Comparing estimators; parametric bootstrap for estimates
- The delta method;
- Random sums; Conditional expectation;
- Continuous distributions and random variables: uniform, exponential, gamma, normal, beta, Cauchy; Chi^2; t; F;
- Functions of random variables (continuous case);
- Independent random variables; Sums and quotients of independent random variables;
- Transformation of bivariate RVs;
- Expectation, variance and covariance (continuous case);
- Bivariate normal distribution;
- Simulations;
- Order Statistics;
- Estimation of parameters using Method of Moments and Maximum Likelihood (continuous case);
- Q-Q plots; probability plots;
- Random sums and Conditional expectation (continuous case);
- Central limit theorem and applications;
- Interval estimation; CI for mean of normal; CI for variance;
- CI for proportions; bootstrap CI
- Distributions related to the normal; Sampling distributions of statistics from normal distributions;

In addition students are expected to demonstrate they are able to make the necessary rigorous arguments that constitute a correct mathematical proof in any of the above subjects. Finally, students are expected to demonstrate their competency in using R for:

- manipulating elementary data structures
- conducting simulation studies
- visualizing features of the data and/or aspects of the analysis
- highlighting salient features of the data
- demonstrating knowledge of basic programming concepts including the use of functions

## Timetable

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