## MAS113 Introduction to Probability and Statistics

 Both semesters, 2017/18 20 Credits Lecturer: Dr Jonathan Jordan Home page (also MOLE) Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

The module provides an introduction to the fields of probability and statistics, which form the basis of much of applicable mathematics and operations research. The theory behind probability and statistics will be introduced, along with examples occurring in such diverse areas as medicine, finance, sport, the environment, law and so on. Some of the computational statistical work will make use of the statistics package R.

There are no prerequisites for this module.

The following modules have this module as a prerequisite:

 MAS223 Statistical Inference and Modelling MAS275 Probability Modelling MAS281 Probability and Statistics in Society MAS352 Stochastic Processes and Finance MAS452 Stochastic Processes and Finance

## Office hours

Tuesdays, 14:30 to 15:30. Alternatively, email me for an appointment or knock on my door if I'm in.

## Aims

• Introduce students to the theory of probability, including applications to practical examples;
• To develop the students' knowledge and understanding of statistics.

## Learning outcomes

• interpret and perform calculations involving random variables and distributions;
• recognise important standard distributions;
• apply the idea of conditional probability via the law of total probability and Bayes' rule;
• use the software package R for simple calculations, handling data, plots, and working with standard distributions;
• calculate standard errors and properties of sampling distributions in simple problems;
• understand the formulation of inference problems in terms of data and model parameters;
• understand the form and logical basis of significance tests, and be able to interpret such tests;
• understand the concept of a confidence interval and the relationship between confidence intervals and tests;
• understand the basis of simple inference procedures for normal expectations and binomial proportions, and be able to use the procedures in R.

## Teaching methods

In both semesters 1 and 2, there will be two formal lectures per week, which will involve the formulation of new theory and worked examples. Each week students will also attend one tutorial, where they will work through set exercises. Areas of common difficulty may be explained on the board by the tutorial leader. Students will also submit homework for marking (but these will not count towards the assessment).

44 lectures, 22 tutorials

## Assessment

A formal, closed book, two hour examination at the end of the second semester (80%), online tests (10%), R practical assignments (10%).

## Full syllabus

1. Introduction
Statistical and probabilistic modelling, and the need for a mathematical theory of chance.

2. Basic Probability
Sets, unions, intersection, complement. Venn diagrams. Sample spaces and events.
The idea of measure of a set. Counting measure. Properties of measures. Probability as measure.
Calculating probabilities in practice - use of symmetry, relative frequencies, subjective probability.
Joint and conditional probability, Bayes theorem, prior and posterior probabilities. Independence.
3. Discrete Random Variables
Discrete random variables. Cumulative distributions and probability laws/ mass functions.
Expectation and variance and their properties (e.g. E(X+Y)=E(X)+E(Y), E(aX+b)=aE(X)+b, Var(aX+b)=a2Var(X).)
Bernoulli, binomial, Poisson and geometric random variables. Calculations of laws, means and variances. The Poisson distribution as the limit of a binomial. The binomial and Poisson distribution in R.
Multivariate discrete random variables. Covariance and correlation between two discrete random variables. The multinomial distribution.
4. Continuous Random Variables
Area under a curve as a measure. Probability via integration. Continuous random variables and their pdfs.
Examples. Uniform and exponential distributions.
Mean and variance as integrals.
The normal distribution. The normal distribution in R. The standard normal Z. Mean and variance in general case via X = σZ + μ.
6. Introducing Statistics
Independent random variables. Sums of i.i.d. random variables. Chebyshev's inequality and the law of large numbers.
The central limit theorem.
The expectation and variance of barX.
7. Summarising and plotting data using R
Working with data in R; calculating summary statistics; plots for visualising data.
8. Introduction to statistical inference
Probabilistic modelling of random sampling from a population. Estimating parameters of probability distributions.
9. Point estimation
Estimating a mean, a proportion and a variance. Expectation and variance of estimators.
10. Interval estimation
Confidence interval for the mean with known and unknown variance, large samples and use of the CLT. Confidence intervals for proportions.
11. Hypothesis tests
Fisher's p-value method, and Neyman-Pearson testing. The t test (one and two sample). Implementation using R. The power of a test, and choosing a sample size. Implementation in R.
12 Analysis of contingency tables
The χ2 test for the analysis of contingency tables. Implementation in R.

Type Author(s) Title Library Blackwells Amazon
A Applebaum, David Probability and information : an integrated approach (2nd ed) 519 (A) Blackwells Amazon
A Dekking, FM, Kraaikamp, C, Lopuhaa, HP and Meester, LE A modern introduction to probability and statistics: understanding why and how 519.2 (D) Blackwells Amazon
A Ross, Sheldon M. A first course in probability (8th ed) 519.2 (R) Blackwells Amazon
A Trosset, Michael W. An introduction to statistical inference and its applications with R 519.50285 (T) Blackwells Amazon
B Grimmett, Geoffrey, and Welsh, Dominic Probability : an introduction 519.2 (G) Blackwells Amazon
C Blastland, Michael and Dilnot, Andrew W. The tiger that isn't : seeing through a world of numbers 510 (B) Blackwells Amazon
C Pruim, Randall Foundations and Applications of Statistics Blackwells Amazon
C Schoenberg, Frederic P. Introduction to probability with Texas hold'em examples 519.2 (S) Blackwells Amazon
C Silver, Nate The Signal and the Noise: The Art and Science of Prediction 303.49 (S) Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 1)

 Mon 09:00 - 09:50 lecture Alfred Denny Building Lecture Theatre 2 Tue 12:00 - 12:50 lecture Dainton Building Lecture Theatre 1 Wed 09:00 - 09:50 lab session (group 80) Arts Tower Computer Room 1012 Wed 09:00 - 09:50 lab session (group 81) Bartolome House Computer Room ALG04 Wed 09:00 - 09:50 lab session (group 82) Diamond Computer Room 4 / Room 206 Wed 09:00 - 09:50 tutorial (group 90) Hicks Seminar Room F20 Wed 09:00 - 09:50 tutorial (group 91) Hicks Seminar Room F24 Wed 09:00 - 09:50 tutorial (group 92) Hicks Seminar Room F28 Wed 09:00 - 09:50 tutorial (group 93) Arts Tower Lecture Theatre 1 Wed 11:00 - 11:50 lab session (group 83) Arts Tower Computer Room 1012 Wed 11:00 - 11:50 lab session (group 84) Diamond Computer Room 4 / Room 206 Wed 11:00 - 11:50 lab session (group 85) FC-B56 - Firth Court Wed 11:00 - 11:50 tutorial (group 94) Hicks Seminar Room F24 Wed 11:00 - 11:50 tutorial (group 95) Hicks Seminar Room F20