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

#### Module Aims

To present the general theory of statistical distributions as well as the standard distributions found in statistical practice, and the relationships among them, to provide a good grounding in the general theory of statistical distributions and to derive many important statistical distributions using the general theory of the calculus of random variables including the use of the moment generating function.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Determine and interpret independence and conditional distributions
• Recall definitions of probability function, density function, cumulative distribution function and moment generating function, and their inter-relationships
• Calculate moments and moment generating function
• Derive chi-square, t and F distributions from normal distribution
• Use moment generating function to determine distribution function and moments
• Recall well known distributions such as Bernoulli, binomial, Poisson, geometric, uniform, exponential, normal, Cauchy, gamma and beta distributions
• Construct z, chi-square, t and F tests and the corresponding confidence intervals from sample means and sample variances
• Find distributions of functions of random variables, including distributions of maximum and minimum observations

### 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; 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 binomial and Poisson mean and 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

ROSS S A. First Course in Probability.

GRIMMETT G & WELSH D. Probability - An Introduction.

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

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

HOEL P G. Introduction to Mathematical Statistics.

### Assessment

#### Summative

MethodPercentage contribution
Test 20%
Written exam  (2 hours) 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External