## STAT261 Probability and Inference 1

First Semester |

In first year, statistics courses emphasise the methods of statistics: which techniques and tests are applied in which situations. In this course, you will learn some of the theory and mathematics behind those methods. This is important because you will

- better understand where those standard methods come from, and why they are used,
- learn how to conduct analyses and design statistical methods for the many cases where the ‘standard’ toolbox is inadequate.

Modern statistics is a dynamic and rapidly changing subject. If you are going to keep up with the changes and advances in statistical theory and methodology you will need a good grounding in mathematical statistics and probability theory.

### Paper details

The mathematical requirements for this course are kept at first year level, i.e. MATH 160. No previous knowledge of probability (beyond that in STAT 110 and 115) is assumed. There will be lots of examples and practice problems.

### Potential students

Any student who has taken either of the 100-level statistics papers and MATH 160 can take this paper. It is particularly useful for those majoring in mathematics, statistics, economics, finance and quantitative analysis, psychology, zoology, or any other field in which statistics is used.

### Prerequisites

MATH 160 and one of STAT 110, STAT 115, COMO 101, BSNS 112, BSNS 102, QUAN 101

### Main topics

- Introduction to probability
- Random variables and distributions
- Expectation and variance
- Transformations of random variables
- Statistical models
- Estimators and likelihood
- Confidence intervals and hypothesis testing
- Bayesian inference

### Required text

None. Course notes will be available on the Resources part of the STAT 261 webpage.

### References

The following two texts are available as electronic e-books in the library:

*A Modern Introduction to Probability and Statistics: Understanding Why and How* by Dekking, Kraaikamp, Lopuhaa and Meester

*Modern Mathematical Statistics with Applications* by Devore and Berk

Additional reading:

*Mathematical Statistics with Applications* by Wackerly, Mendenhall and Scheaffer

*An Introduction to Mathematical Statistics and its Applications* by Larsen and Marx

### Lecturers

Dr Matthew Parry, Room 236

Dr Ting Wang, Room 518

### Lectures

Tuesdays, Thursdays and Fridays at 10 am in Room 241, Science III

### Tutorials

Wednesdays at 3-5pm in Laboratory B21, Science III

### Internal Assessment

There will be 10 assignments, each of which contributes 6% of the internal assessment mark, and a mid-term test that contributes the remaining 40%.

### Exam

There will be a three-hour exam

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, (2E + 0.6A + 0.4T)/3)**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the Test mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

### Jakob Bernoulli

One of a Swiss family producing eight distinguished scientists, Jakob Bernoulli (1654-1705) was forced by his father to pursue theological studies, but his love of mathematics eventually led him to a university career. Bernoulli’s main work in probability,*Ars Conjectandi*, was published after his death by his nephew, Nikolaus in 1713.

Modern probability theory had its start in 1654 when the French nobleman Chevalier de Mere wrote a letter to Blaise Pascal, a celebrated mathematician, to discuss the following gambling problems.

De Mere thought these two events were equally likely. For the first game, he reasoned it as follows: in one roll of a die, I have 1/6 of a chance to get an ace, so in 4 rolls, I have 4 x 1/6 of a chance to get at least one ace.

His reasoning for the second game was similar: in one roll of a pair of dice, I have 1/36 of a chance to get at least one ace, so in 24 rolls, I must have 24 x 1/36 =2/3 of a chance to get at least one double-ace.

Using this faulty argument, both chances were the same, namely 2/3. However experience showed the first event to be a bit more likely than the second. This contradiction became known as the Paradox of the Chevalier de Mere.

What are the probabilities of those two events?

### Marquis de Laplace

Marquis de Laplace (1749-1827) wrote the book,*Analytical Theory in Probability*, where he presented 10 principles of probability calculations as a general introduction and then he went on to apply these to natural philosophy and moral sciences. The laws or theorems of probability calculations have not changed since Laplace’s time.