## MAS5051 Probability and Probability Distributions

Both semesters, 2021/22 | 20 Credits | ||||

Lecturer: | Dr Mark Yarrow | Reading List | |||

Outcomes | Teaching Methods | Assessment | Full Syllabus |

The module is a distance-learning instrument aimed at graduates who wish to undertake a postgraduate Masters course in Statistics but lack the appropriate mathematical and statistical background. It introduces the concepts of probability theory and distribution theory needed for such a masters course, and gives training in their use. It covers the laws of probability and of conditional probability, the concepts of random variables and random vectors and their distributions, and the methodology for calculating with them. It also discusses Laws of Large Numbers and Central Limit phenomena.

There are no prerequisites for this module.

No other modules have this module as a prerequisite.

## Learning outcomes

By the end of the unit, students should be able to demonstrate: - A knowledge of the concepts of probability theory and distribution theory, and an understanding of their use. - A knowledge of the laws of probability and of conditional probability; - A knowledge of the concepts of random variables and random vectors and their distributions, and the methodology for calculating with them. - A knowledge of the Laws of Large Numbers and Central Limit phenomena.## Teaching methods

Directed reading from the textbook listed below; discussion boards; coursework assignments.

No lectures, no tutorials

## Assessment

Formal Examination, Coursework

## Full syllabus

- Sample spaces and events
- Defining probabilities
- Conditional probability
- Independence
- Discrete random variables
- Expectation (mean), variance and standard deviation
- Independence of random variables
- Mean and variance of linear combinations
- Bernoulli trials and the Binomial distribution
- The Geometric distribution
- Continuous random variables
- Mean and variance for continuous random variables
- Moment generating functions
- The normal distribution
- The Poisson distribution
- The Gamma and exponential distributions
- The Beta distribution
- Transformations of random variables
- The chi-squared, t and F distributions
- Distributions in R
- Normal approximations to Binomial and Poisson
- The Weak Law of Large Numbers
- The Central Limit Theorem
- Bayes' Theorem
- Multivariate random variables
- The Multinomial Distribution
- Multivariate continuous random variables and conditional distributions
- Covariance and correlation
- Conditional expectation
- Multivariate normal distribution
- Transformations of multivariate random variables
- Examples and applications of multivariate transformations
- Introduction to Markov Chains
- Basic Bayesian inference

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

A |
Miller, I. and Miller, M. | John E. Freund's Mathematical Statistics with Applications (8th edition) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.