The University of Southampton
Courses

# MATH2011 Statistical Distribution Theory

## Module Overview

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

### Aims and Objectives

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

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

### Syllabus

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.

TypeHours
Independent Study102
Teaching48
Total study time150

MOOD A M, GRAYBILL F A & BOES D C. Introduction to the Theory of Statistics.

ROSS S A. First Course in Probability.

HOEL P G. Introduction to Mathematical Statistics.

FARAWAY J.J (2005). Linear Models with R.

GRIMMETT G & WELSH D. Probability - An Introduction.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 30%
Written assessment 70%

#### Referral

MethodPercentage contribution
Written assessment 100%

#### Repeat Information

Repeat type: Internal & External

Prerequisites: MATH1024 and MATH1059 and MATH1060

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Recommended texts for this module may be available in limited supply in the University Library and students may wish to purchase reading texts as appropriate.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.