## MAS113 Introduction to Probability and Statistics

Both semesters, 2021/22 | 20 Credits | ||||

Lecturer: | Dr Jill Johnson | 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 |

MAS286 | Mathematics and Statistics in Action |

MAS352 | Stochastic Processes and Finance |

MAS452 | Stochastic Processes and Finance |

## Office hours

## 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

Three separate assessments, **all** of which must be passed to pass the module:

- exam;
- probability online tests;
- statistics practical assignments.

## 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)=a

^{2}Var(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. Independent and identically distributed random variables, and the central limit theorem**

Independent random variables. Sums of i.i.d. random variables; expectation and variance. Chebyshev's inequality and the law of large numbers. Moment generating functions. The central limit theorem.

**7. Summarising and plotting data using R**

Working with data in R using the "tidyverse"; calculating summary statistics; plots for visualising data.

**8. A short introduction to machine learning**

Classification with the nearest-neighbour algorithm.

**8. Introduction to statistical modelling**

Using probability distributions to model data; parameters of probability distributions as population characteristics.

**9. Point estimation**

Estimating a mean, a proportion and a variance. Unbiased estimators. Standard errors. Consistency.

**10. Interval estimation**

Confidence intervals for means, variances and proportions.

**11. Hypothesis tests**

Fisher's p-value method and Neyman-Pearson testing. Two sample t-test. Using simulation to understand hypothesis tests. Comparing two binomial proportions. χ

^{2}test for the analysis of contingency tables. Power and sample size considerations. Implementation in R.

## Reading list

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 | |||

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.