## Handbook home

# Probability and Random Models (ELEN90054)

Graduate courseworkPoints: 12.5On Campus (Parkville)

## Overview

Availability | Semester 1 |
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Fees | Look up fees |

**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:

- Demonstrate an understanding of combinatorics, the axioms of probability, independence, random variables, conditioning and Bayes’ rule
- Demonstrate an understanding of important distributions, stochastic processes and decision tests, and their significance
- Formulate random models of signals and systems encountered in engineering
- Calculate and interpret probabilities, probability densities, means, variances and covariances, from given information
- Use the law of large numbers, the central limit theorem, and inequalities to find approximations and bounds
- 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: 23 April 2024