Statistics
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Department of Mathematics & Statistics

## STAT261 Probability and Inference 1

 First Semester
18 points
Not available after 2018

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

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

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.
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.

In the 17th century French gamblers used to bet on the event that in 4 rolls of a die, at least one ace would turn up; an ace is a one. In another game they bet on the event that in 24 rolls of a pair of dice, at least one double-ace would turn up.

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 (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.
Karl Pearson (1857-1936) has been called “the founder of the science of statistics”. Pearson’s research involved the laws of heredity, but to carry out his investigations required the development and extension of statistical methods. These included the fitting of mathematical curves to the frequency distributions of observed data, the development of basic formulae in simple and multiple correlation, and the introduction of a chi-square test of goodness-of-fit of a mathematical curve, or model, to observed data.