# Probability and Random Models (ELEN90054)

Graduate courseworkPoints: 12.5On Campus (Parkville)

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## Overview

Year of offer 2019 Graduate coursework ELEN90054 Parkville Semester 1Semester 2 Subject EFTSL, Level, Discipline & Census Date

AIMS

This subject provides an introduction to probability theory, random variables, random vectors, decision tests, and stochastic processes. Uncertainty is inevitable in real engineering systems, and the laws of probability offer a powerful way to evaluate uncertainty, to predict and to make decisions according to well-defined, quantitative principles. The material covered is important in fields such as communications, data networks, signal processing and electronics. This subject is a core requirement in the Master of Engineering (Electrical, Mechanical and Mechatronics).

INDICATIVE CONTENT

Topics include:

• Foundations – combinatorial analysis, axioms of probability, independence, conditional probability, Bayes’ rule;
• Random variables (rv’s)– definition; cumulative distribution, probability mass and probability density functions; expectation and variance; functions of an rv; important distributions and their properties and uses;
• Multiple random variables – joint cumulative distribution, probability mass and probability density functions; independent rv’s; correlation and covariance; conditional distributions and expectation; functions of several rv’s; jointly Gaussian rv’s; random vectors;
• Sums, inequalities and limit theorems – sums of rv’s, moment generating function; Markov and Chebychev inequalities; weak and strong laws of large numbers; the Central Limit Theorem;
• Decision testing - maximum likelihood, maximum a posterior, minimum cost and Neyman-Pearson rules; basic minimum mean-square error estimation;
• Stochastic processes – mean and autocorrelation functions, strict and wide-sense stationarity; ergodicity; important processes and their properties and uses;
• Introduction to Markov chains.

This material is complemented by exposure to examples from electrical engineering and software tools (e.g. MATLAB) for computation and simulations.

## Intended learning outcomes

INTENDED LEARNING OUTCOMES (ILOs)

Having completed this subject it is expected that the student be able to:

1. Demonstrate an understanding of combinatorics, the axioms of probability, independence, random variables, conditioning and Bayes’ rule
2. Demonstrate an understanding of important distributions, stochastic processes and decision tests, and their significance
3. Formulate random models of signals and systems encountered in engineering
4. Calculate and interpret probabilities, probability densities, means, variances and covariances, from given information
5. Use the law of large numbers, the central limit theorem, and inequalities to find approximations and bounds
6. Simulate random models using software tools

## Generic skills

On completion of this subject, students will have developed the following skills:

• Ability to apply knowledge of basic science and engineering fundamentals;
• In-depth technical competence in at least one engineering discipline;
• Ability to undertake problem identification, formulation and solution;
• Ability to utilise a systems approach to design and operational performance;
• Capacity for independent critical thought, rational inquiry and self-directed learning;
• Ability to communicate effectively, with the engineering team and with the community at large.

Last updated: 3 April 2019