Unit information: Probability 1 in 2012/13

Unit name	Probability 1
Unit code	MATH11300
Credit points	10
Level of study	C/4
Teaching block(s)	Teaching Block 1 (weeks 1 - 12)
Unit director	Professor. McNamara
Open unit status	Not open
Pre-requisites	A good pass in A-level Maths or equivalent
Co-requisites	MATH11007 Calculus 1
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

This unit introduces the basic ideas and methods of Probability. Some familiarity with calculus is needed. Probability is an everyday concept of which most people have only a vague intuitive understanding. Study of games of chance resulted in early attempts to formalise the theory; but a satisfactory rigorous basis for the subject only came with the axiomatic theory of Kolmogorov in 1933. The unit starts with the idea of a probability space, which is how we model the outcome of a random experiment. Probability models are then introduced in terms of random variables (which are functions of the outcomes of a random experiment), and the simpler properties of standard discrete and continuous random variables are discussed. Motivation is given for studying the common quantities of interest (probabilities, expected values, variances and covariances). Techniques are developed for evaluating these quantities, including generating functions, conditional expectations and simple approximate methods.

Aims:

To introduce the basic ideas and methods of Probability, developing the concepts of random variables, expectations and variances. To look at some simple applications of these ideas and methods.

Syllabus

• Sample spaces; events; axioms of probability; simple results derived from the axioms [3].

• Combinatorial probability [2].

• Independent events; conditional probability; multiplications lemma; partition theorem; Bayes theorem [2].

• Discrete random variables (r.v.'s); probability mass function; Bernoulli, binomial, Poisson and geometric distribution; Poisson approximation to binomial [2].

• Expectations of r.v.'s; expectations of a function of r.v.'s; variance of r.v.'s and standard deviation [2].

• Continuous random variables; distribution function; probability density function; uniform, exponential, gamma and normal distributions; use of statistical tables; transformations [4].

• Bivariate distributions; joint, conditional and marginal distributions; independent random variables [1].

• Properties of expectations(linear combinations, independent products) [2].

• Properties of variance and covariance (linear combinations, sums of independent r.v.'s, degenerate r.v.'s); correlation [2].

• Moment generating functions ; moments; linear combinations; applications to normal distributions; independent r.v.'s [2].

• Conditional expectation; partition theorem; formulae for E(X) in terms of E(X|Y); random sums [2].

Relation to Other Units

This unit provides the foundation for all probability and statistics units in later years.

Intended Learning Outcomes

When you have successfully completed this module you will be able to:

Define events and sample spaces, describe them in simple examples, and use counting arguments to calculate probabilities when there are equally likely outcomes.

• List the axioms of probability, and use them to prove simple results, including the partition theorem and Bayes’ theorem.

• Define a random variable. Define the probability mass function for discrete random variables, and the probability density function (pdf) and cumulative distribution function (cdf) for continuous random variables. Illustrate links between the pdf and cdf. Calculate the pdf of a function of a random variable.

• Define the following random variables: Bernoulli, binomial, geometric, Poisson, uniform, exponential, gamma, normal/Gaussian. Recall and illustrate features of these distributions.

• Define and calculate the expectation, variance and covariance of simple random variables, including all of the standard types in the previous objective.

• Define jointly distributed random variables, joint probability mass functions, and joint probability density functions.

• Define the moment generating function of a random variable. Use moment generating functions to analyse sums of random variables.

• Define and explain conditional expectation. Prove the double expectation formula. Use conditional expectation and moment generating functions to analyse random sums.

• Formulate formal probability models from informal descriptions.

Transferable Skills:

Model building. Especially the formal mathematical modelling of informal descriptions of events and processes.

Teaching Information

Lectures supplemented (for first year students) by small group tutorials. Weekly problem sheets, with outline solutions handed out a fortnight later.

Assessment Information

The final mark for Probability 1 is calculated from one 1½ -hour written examination in April. This examination paper is in two sections.

• Section A contains 5 short questions, ALL of which should be attempted. Section A contributes 40% of the mark for this paper.

• Section B has 3 longer questions; you should attempt TWO. If you attempt more than two, your best two answers in Section B will be used for assessment. Section B contributes 60% to the mark for this paper.

Calculators are not permitted in the examination.

September examinations

If you fail Probability 1 (or any other unit in the Science Faculty), you may be required to resit in September. Your departmental or Faculty handbook explains the conditions under which resits may be allowed. The September examinations have the same format as the April examination (given above).

Reading and References

The recommended text is:

S. M. Ross A First Course in Probability, Prentice Hall International A suitably abbreviated version of this text, at a reduced price, will be available.

The statistical software package R will be used during the course. This software is available on the computers in the undergraduate laboratory, and for home use is available to download for free from the R project homepage.