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Jakob Bernoulli (1654-1705)

One of a Swiss family producing eight distinguished scientists, Jakob 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.

STAT261 Probability and Inference 1

First Semester, 18 points
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;
  • You will 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 which statistics can be used to solve real life problems or to carry out a scientific investigation.

Main topics

  • Introduction to probability
  • Random variables and distributions
  • Simulation
  • Expectation and variance
  • Asymptotic results
  • Exploratory data analysis
  • Statistical models
  • Estimation
  • Likelihood
  • Hypothesis tests and confidence intervals

Prerequisites

STAT 110 or STAT 115, and MATH 160

Required text

Dekking, F.M., Kraaikamp, C., Lopuhaa, H.P. and Meester, L.E. A Modern Introduction to Probability and Statistics: Understanding Why and How. Springer.

An e-book of this text is available free for all Otago students from the University Library.

References

Additional reading:
Mathematical Statistics with Applications by Wackerly, Mendenhall and Scheaffer, 7ed.
An Introduction to Mathematical Statistics and its Applications by Larsen and Marx.

Lecturers

Assoc Prof David Bryant, room 514 and Dr Ting Wang, room 518

Lectures

Tuesday, Thursday and Friday at 10 am. (total of 32 lectures)

Tutorials

TBA

Internal Assessment

TBA

Exam format

TBA

Final mark


TBA
The final mark F is calculated from:
F = E + A
where E (exam mark) is out of 60, A (internal assessment) is out of 40.


Plagiarism

Students should make sure that all submitted work is their own. “Plagiarism is a form of dishonest practice. Plagiarism is defined as copying or paraphrasing another’s work and presenting it as one’s own” (University of Otago Calendar). In practice this means that plagiarism includes any attempt in any piece of submitted work (e.g. an assignment or test) to present as one’s own work the work of another (whether of another student or a published authority). Any student found to be responsible for plagiarism in any piece of work submitted for assessment shall be subject to the University’s dishonest practice regulations which may result in various penalties, including forfeiture of marks for the piece of work submitted, a zero grade for the paper, or in extreme cases exclusion from the University. The University of Otago reserves the right to use plagiarism detection tools.

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.