Statistical Distribution Theory

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• 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
• 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
• Calculate moments and generating functions
• Use generating functions to determine distribution function and moments
• Derive chi-squared, t and F distributions from normal distribution
• Recall well known distributions such as Bernoulli, binomial, Poisson, geometric, uniform, exponential, normal, Cauchy, gamma and beta distributions
• Recall definitions of probability function, density function, cumulative distribution function and generating functions, and their inter-relationships
• Determine and interpret independence and conditional distributions

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.

Lectures, in class tests, problem classes, private study.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.