Functions of one and several random variables are considered such as sums, differences, products and ratios. The central limit theorem is proved and the probability density functions are derived of those sampling distributions linked to the normal distribution. Bivariate and multivariate distributions are considered, and distributions of maximum and minimum observations are derived.
This module is a pre-requisite for all subsequent statistics modules, and desirable for Actuarial Mathematics I and II and Simulation and Queues
Prerequisites: MATH1024 and MATH1059 and MATH1060
Aims and Objectives
Having successfully completed this module you will be able to:
- Determine and interpret independence and conditional distributions
- Calculate moments and generating functions
- Recall well known distributions such as Bernoulli, binomial, Poisson, geometric, uniform, exponential, normal, Cauchy, gamma and beta distributions
- Find distributions of functions of random variables, including distributions of maximum and minimum observations, and use these results to derive methods to simulate observations from standard distributions
- Derive chi-squared, t and F distributions from normal distribution
- Use generating functions to determine distribution function and moments
- Construct z, chi-squared, t and F tests and the corresponding confidence intervals from sample means and sample variances, and apply chi-squared tests for contingency tables and goodness of fit
- Recall definitions of probability function, density function, cumulative distribution function and generating functions, and their inter-relationships
Random variables; probability, probability density and cumulative distribution functions; Expected value and variance of a random variable. Bernouilli trials, binomial, Poisson, geometric, hypergeometric, negative binomial distributions, and their inter-relationships. Poisson process. Probability generating functions.
Moment and cumulant generating functions; exponential, gamma, normal, lognormal, uniform, Cauchy and beta distributions.
Joint distributions; conditional distributions; independence; conditional expectations. Covariance, correlation.
Distributions of functions of random variables, including sums, means, products and ratios. Transformations of random variables; simulating observations from standard distributions; use of Jacobians; marginal distributions.
Proof of Central Limit Theorem. Derivation of chi-squared, t and F distribtions, and their uses. Distributions of sample mean and sample variance.
Estimation: Method of moments and maximum likelihood, efficiency, bias consistency and mean square error, unbiasedness, asymptotic properties of estimators. Confidence intervals for one and two samples for means and variances of normal distributions. Use of paired data.
Introduction to statistical inference; hypothesis testing; significance level, power, likelihood ratios, particularly demonstrating uses of chi-squared, t and F distributions, Bayesian inference.
Multivariate distributions and moment generating function; multinomial distribution; bivariate normal distribution; correlation. Distributions of maximum and minimum observations.
Compound distributions: conditional expectations, mean and variance of a random variable from expected values of conditional expected values.
Learning and Teaching
Teaching and learning methods
Lectures, in class tests, problem classes, private study.
|Total study time||150|
Resources & Reading list
FARAWAY J.J (2005). Linear Models with R. Chapman and Hall/CRC.
GRIMMETT G & WELSH D. Probability - An Introduction. Oxford.
ROSS S A. First Course in Probability. Collier-MacMillan.
MOOD A M, GRAYBILL F A & BOES D C. Introduction to the Theory of Statistics. McGraw-Hill.
HOEL P G. Introduction to Mathematical Statistics. Wiley.
Summative assessment description
Referral assessment description
Repeat type: Internal & External