# Mathematical Statistics - MAST5007

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## Module delivery information

Location Term Level1 Credits (ECTS)2 Current Convenor3 2023 to 2024
Canterbury
Autumn Term 5 15 (7.5) Alexa Laurence

## Overview

Probability: Joint distributions of two or more discrete or continuous random variables. Marginal and conditional distributions. Independence. Properties of expectation, variance, covariance and correlation. Poisson process and its application. Sums of random variables with a random number of terms.

Transformations of random variables: Various methods for obtaining the distribution of a function of a random variable —method of distribution functions, method of transformations, method of generating functions. Method of transformations for several variables. Convolutions. Approximate method for transformations.

Sampling distributions: Sampling distributions related to the Normal distribution — distribution of sample mean and sample variance; independence of sample mean and variance; the t distribution in one- and two-sample problems.

Statistical inference: Basic ideas of inference — point and interval estimation, hypothesis testing.

Point estimation: Methods of comparing estimators — bias, variance, mean square error, consistency, efficiency. Method of moments estimation. The likelihood and log-likelihood functions. Maximum likelihood estimation.

Hypothesis testing: Basic ideas of hypothesis testing — null and alternative hypotheses; simple and composite hypotheses; one and two-sided alternatives; critical regions; types of error; size and power. Neyman-Pearson lemma. Simple null hypothesis versus composite alternative. Power functions. Locally and uniformly most powerful tests.
Composite null hypotheses. The maximum likelihood ratio test.

Interval estimation: Confidence limits and intervals. Intervals related to sampling from the Normal distribution. The method of pivotal functions. Confidence intervals based on the large sample distribution of the maximum likelihood estimator – Fisher information, Cramer-Rao lower bound. Relationship with hypothesis tests. Likelihood-based intervals.

## Details

### Contact hours

Total contact hours: 44
Private study hours: 106
Total study hours: 150

## Method of assessment

Assessment 1 Exercises, requiring on average between 10 and 15 hours to complete 15%
Assessment 2 Exercises, requiring on average between 10 and 15 hours to complete 15%
Examination 2 hours 70%
The coursework mark alone will not be sufficient to demonstrate the student's level of achievement on the module.

MILLER, I. and MILLER, M. (2014) John E. Freund's Mathematical Statistics with Applications. 8th international edition. Pearson Education, Prentice Hall, New Jersey.
LINDLEY, D.V. and SCOTT, W.F. (1995) New Cambridge Statistical Tables. 2nd edition.
HOGG, R., CRAIG, A. and McKEAN, J. (2003) Introduction to Mathematical Statistics. 6th international edition.
LARSON, H. J. (1982) Introduction to Probability Theory and Statistical Inference. 3rd edition.
SPIEGEL, M. R, SCHILLER, J. and ALU SRINIVASAN, R. (2013) Schaum's Outline of Probability and Statistics. 4th edition.
LEE, P. M. (2012) [for level 6 students] Bayesian Statistics an Introduction. 4th edition. (ebook)

See the library reading list for this module (Canterbury)

## Learning outcomes

The intended subject specific learning outcomes. On successfully completing the level 5 module students will be able to:

1 demonstrate knowledge and critical understanding of the well-established principles within probability and inference;
2 demonstrate the capability to use a range of established techniques and a reasonable level of skill in calculation and manipulation of the material to solve problems in the
following areas: joint, marginal and conditional probability distributions, to derive distributions of transformed random variables, to calculate point and interval estimates of
parameters and to perform tests of hypotheses;
3 apply the concepts and principles in probability and inference in well-defined contexts beyond those in which they were first studied, showing the ability to evaluate
critically the appropriateness of different tools and techniques.

## Notes

1. Credit level 5. Intermediate level module usually taken in Stage 2 of an undergraduate degree.
2. ECTS credits are recognised throughout the EU and allow you to transfer credit easily from one university to another.
3. The named convenor is the convenor for the current academic session.