# Mathematics

## ECM1707 - Probability and Discrete Mathematics (2015)

MODULE TITLE CREDIT VALUE Probability and Discrete Mathematics 15 ECM1707 Dr Robin Chapman (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 260
DESCRIPTION - summary of the module content

Discrete mathematics is concerned with quantities, which vary discretely, as opposed to the continuous variables you have encountered in calculus.  Therefore, on this module, you will be concerned with counting rather than measuring. For example, you will learn how to enumerate permutations and combinations of objects satisfying specified conditions. This example provides a link to probability theory, where we determine the probability of each outcome of a combinatorial experiment by expressing the number of different ways that outcome can be realised as a fraction of all the possibilities. But probability also uses continuous mathematics for cases where the possible outcomes of an experiment form a continuous range. This module will provide you with the basic concepts and tools needed for the study of discrete mathematics and probability. As such, it forms an essential part of a rounded mathematical education, and in particular it is a prerequisite for the modules in the statistics stream within the mathematics syllabus.

Prerequisite module: ECM1702 and  ECM1705 or equivalent

AIMS - intentions of the module

The aim of this module is to introduce you to some basic topics in discrete mathematics and to the elementary concepts of probability, including random variables and common probability distributions. In general, the module aims to provide you with some basic, essential knowledge relevant to all mathematical studies. In particular, it aims to provide a foundation for further study in statistics or stochastic processes.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate a sound understanding of selected essential topics in discrete mathematics and of basic probability theory, including the ability to apply those concepts in tackling an appropriate range of problems.
Discipline Specific Skills and Knowledge:
2 show sufficient knowledge of fundamental mathematical concepts, manipulations and results, including relevant finite and infinite summations.
Personal and Key Transferable/ Employment Skills and  Knowledge:
3 reason using abstract ideas, formulate and solve problems and communicate reasoning and solutions effectively in writing;
4 use learning resources appropriately;
5 exhibit self management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- review of set theory and standard notation;

- review of sums of standard finite and infinite series;

- counting principles;

- methods of proof;

- probability;

- random variables;

- discrete probability distributions;

- discrete bivariate and multivariate distributions;

- continuous random variables: the exponential, gamma, uniform and normal distributions, expectations and moments;

- continuous bivariate and multivariate distributions: joint, marginal and conditional distributions, conditional expectation and variance, covariance and independence;

- distributions of functions of random variables, linear combinations of random variables, the central limit theorem, approximations to the binomial and Poisson distributions; distributions derived from the normal distribution.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 49 101 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures Scheduled learning and teaching activities 5 Problem class Scheduled learning and teaching activities 11 Tutorials Guided independent study 10 Coursework Guided independent study 91 Guided independent study

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Fortnightly exercise 10 hours 1-4 Tutorial

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours 1-4 In accordance with CEMPS policy
Coursework – based on questions submitted for formative assessment 20 10 hours 1-5 Tutorial/seminar/written

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk