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Probability 1 (MATH11300)

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Administrative Information

  1. Unit number and title: MATH11300 Probability 1
  2. Level: C/4 (Honours)
  3. Credit point value: 10 credit points
  4. Year: 08/09
  5. First Given in this form: 07/08 in this form. Previously a component of Prob and Stats 1
  6. Unit Organiser: David Leslie
  7. Lecturer: David Leslie
  8. Teaching block: 1
  9. Prerequisites: An A in A-level Mathematics or equivalent. Corequisites: Analysis 1 (MATH11006) and Calculus 1 (MATH11007), or equivalent.

Unit aims

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

General Description of the Unit

Probability is an everyday concept of which most people have only a vague intuitive understanding. Study of games of chance, such as tossing dice and card games, 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). Finally techniques are developed for evaluating these quantities, including generating functions and conditional expectations.

Relation to Other Units

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

Teaching Methods

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

Learning Objectives

In the probability component, for each of the areas listed below the student should gain an understanding of the underlying theory and an ability to carry out relevant calculations and apply standard methods in practice;

  • Simple probability models, particularly models with equally likely outcomes;
  • Standard discrete and continuous probability distributions;
  • Means, variances and covariances for (sums of) simple random variables;
  • Moment generating functions.
  • Conditional expectations

Assessment Methods

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 of the approved type are required for 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).

 

Award of Credit Points

You gain the credit points for the unit if

  • EITHER your final assessment mark is at least 40 on the standard Science Faculty scale,
  • OR your mark lies in the interval [30, 39] and
    • first-year students must satisfy the conditions on handing-in of homework and attendance at tutorials described in the First Year Handbook.
    • Students in their second, third or fourth years must have handed in attempts to 75% of the homework and attended the January exam.

Transferable Skills

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

Texts

The recommended text is: S. M. Ross A First Course in Probability, Prentice Hall International.

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.

Syllabus

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].