MATH20701 - 2008/2009

 

Unit specification

Aims

The programme unit aims to develop a solid foundation in the calculus of probabilities and indicate the relevance and importance of this to tackling real-life problems.

Brief description

This course continues the development of probability and statistics from the first year so that all students on the single honours programme have the basic grounding in this area which would be expected of a mathematics graduate. It provides a solid basis for a wide variety of options later in the programme for students who wish to take their studies in probability and/or statistics further.

Intended learning outcomes

On completion of this unit successful students will be able to:

• understand the concept of both univariate and multivariate random variables;
• evaluate the distribution of functions of random variables and calculate expectations;
• understand the concepts of confidence intervals and hypothesis testing

Future topics requiring this course unit

All course units in Probability and Statistics.

Syllabus

1. Very quick review of random variables, distribution functions, p.m.f. and p.d.f. Bivariate distributions, marginal distributions, conditional distributions. [3 lectures]
2. Expectations of functions of bivariate random variables. Correlation and covariance of two random variables. Conditional expectations. [3]
3. Independence of two random variables. Multivariate distributions and independence of more than two random variables. The bivariate normal distribution. [2]
4. Distribution of a function of random variables. Equivalent event method and use of distribution functions. Bivariate transformations. [4]
5. Sums and linear forms of random variables. The means and variances. Covariance between two linear forms of random variables. Distribution of sum and linear forms of random variables. Convolutions. Distribution of linear forms of Normal random variables. Sum of Binomial random variables. Central Limit Theorem. Law of large numbers. [5]
6. Populations. Samples. Sampling distributions. Applications to confidence intervals and hypothesis testing. [5]

Textbooks

• S. Ross, A First Course in Probability, 4th edition, Macmillan.
• D. Stirzaker, Elementary Probability, Cambridge University Press. Available electronically
• Mood, A. M., Graybill, F. A. and Boes, D. C., Introduction to the Theory of Statistics, 3rd edition, McGraw-Hill 1974
• Neil A. Weiss, A Course in Probability, Pearson.

Learning and teaching processes

Two lectures and one examples class each week.

Assessment

Coursework; Weighting within unit 20%

2 hours end of semester examination; Weighting within unit 80%

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Arrangements