Introduction to Probability & Statistics

Overview

This is a fundamental module which provides an introduction to probability theory and the key concepts found in statistics. The topics covered include the laws of probability, discrete and continuous random variables, standard discrete and continuous distributions, bivariate distributions, statistical models, sampling, estimation, hypothesis testing and statistical quality control.

Learning Objectives

- Demonstrate an understanding of the concepts of probability, conditional probability, multiplicative law, independence, Bayes theorem and their interpretations.
- Be able to apply set theory to the proof and use of the axioms of probability.
- Understand and use combinatorial methods: counting rules; sampling with and without replacement; ordered and non-ordered samples.
- Be able to define discrete and continuous random variables and the corresponding probability distributions, probability functions, cumulative distribution functions and probability density functions.
- Understand and use transformations in the discrete and continuous variable context.
- Be able to define expectation and calculate expected values for the mean and variance of specific discrete and continuous distributions.
- Be able to define, interpret and apply the properties of the expectation and variance operators for discrete and continuous cases.
- Demonstrate an understanding of key discrete and continuous distributions including the specific circumstances when distributions may be applied.
- Demonstrate an ability to use statistical tables and deal with linear combinations of independent normal random variables.
- Be familiar with the Central limit theorem for the approximate distribution of sample mean and be able to utilise this theorem in the approximation of binomial and Poisson distributions.
- Understand and be able to define bivariate distributions, their joint probability (density) functions, cumulative distribution functions, marginal distributions, conditional distributions of discrete and continuous random variables.
- Demonstrate an understanding of independence for bivariate data.
- Be able to define expectation and to calculate expected values: means, variances and covariances, correlation coefficients for bivariate distributions and for linear combinations of random variables.
- Be able to define statistical models, experimental, systematic and random errors; precision and accuracy.
- Be able to describe and utilise the following methods of sampling: accessibility, judgement, quota, sequential, random, systematic, stratified and cluster sampling methods.
- Understand the concept of estimation, the definition of a statistic, sampling distribution, sample estimator, sample estimate and the desirable properties for an estimator.
- Be able to define and calculate an estimate of the population mean and variance from a single sample and from several samples.
- Demonstrate an understanding of and be able to implement the method of moments, maximum likelihood estimation and the method of least squares, in particular, the likelihood function, asymptotic variance, normal equations, and linear regression.
- Understand and be able to define the null and alternative hypotheses; one and two-sided tests; test statistic; critical region, P-value, significance level; type I and type II errors; power function and confidence intervals.
- Know when to apply the correct method for significance testing based on given circumstances.
- Be able to interpret results of a significance test and confidence intervals.
- Demonstrate an understanding and be able to describe non-parametric methods and their advantages and disadvantages.
- Understand, be able to carry out and interpret significance tests, in particular key parametric tests based on the Normal distribution, t-distribution, F-distribution and Chi squared distribution, and key non-parametric tests.
- Know when to apply and how to calculate nonparametric statistics and how to choose the appropriate technique to use for a practical example.

Skills

- Understanding when and how to apply probability theory and reasoning with uncertainty.
- Knowing how to apply estimation approaches and the appropriate technique to use.
- Being able to apply probability theory to a practical example.
- Understanding the principles of hypothesis testing.
- Knowing when to apply the correct method for significance testing.
- Calculating test statistics and being able to use these to draw a conclusion about a null hypothesis.
- Knowing when to apply and how to calculate nonparametric statistics and choosing the appropriate technique to use for a practical example.

Assessment

None.