Mathematics
MA199: Statistics & Probability & Maths Of Finance (15 ECTS)
PAPER I:
Paper I is intended as a first course in probability taken by students studying a degree in which mathematics is to be the main subject throughout that degree as it provides a good foundation to higher level probability and statistics courses.
PAPER II:
Paper II material is intended as a first course in statistics for students studying a degree in which mathematics is to be the main subject throughout that degree as it provides a good foundation to higher level probability and statistics modules. The material in paper I is a prerequisite for this material.
PAPER III:
Simple interest and simple discount, bank discount and negotiable instruments, compound interest and discount, equivalent dates and rates, exponential and logarithm functions, annuities (all types),
perpetuity, capitalization, depreciation, loans (amortization and sinking funds).
Taught in Semester(s) I+II. Examined in Semester(s) I+II.
Workload: 44 hours (36 Lecture hours, 8 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
PAPER 1:
- demonstrate the concepts of systematic and random variation, and that probability is concerned with the construction of mathematical models for random phenomena that are subject to stable relative frequencies; comprehend that probability and (inferential) statistics are opposite scientific processes, and be able to give examples where the former is used to justify statistical inferences made in the real world
- demonstrate the role of probability both as a discipline in its own right with applications to e.g. financial decision-making, gambling, communications systems, and as the tool used in justifying statistical inferences (i.e. in justifying statements made about entire populations based on information available in samples taken from the populations)
- demonstrate the frequentist and classical approaches to probability, be able to calculate probabilities for compound events, understand the ideas of mutually exclusive events and of independent events, and be able to perform calculations involving Bayes' formula
- demonstrate the motivation for the introduction of the concept of random variable, and the idea that a given population can be viewed as synonymous with the distribution of an suitably-defined random variable
- model basic discrete random variables and perform calculations based on hypergometric, multivariate hypergeometric, binomial, geometric, negative binomial and Poisson distributions
- demonstrate the importance of the first two moments of discrete and continuous random variables as summary measures of a distribution, and be able to compute the mean and variance of certain discrete variables
- demonstrate the idea underlying the density of a continuous random variable and be able to perform probability calculations for normally distributed variables
- demonstrate the importance and properties of sampling distributions, especially that of the sample mean; be able to calculate probabilities about the mean of a random sample when sampling from a normal distribution
- state the central Limit Theorem and apply it to compute probabilities relating to sums and means of values of both quantitative and Bernoulli variables
PAPER II:
- identify sources of variation in observational and experimental data, identify ideas involved in some basic survey and experimental designs, and be aware of sensitivity of analyses to various assumptions
- summarise data numerically and graphically
- demonstrate how probability is used in the construction of interval estimates and in hypothesis testing, including the computation of p-value and power of tests
- identify and perform some one and two-sample statistical inference procedures for parametric models
- perform basic enumerative data analysis concluding good-of-fit and contingency table tests and tests for equality of several population proportions
- calculate and interpret correlation and conduct analysis for simple linear regression models
PAPER III:
- Calculate: simple and compound interest, simple and compound discount, bank discount, present values, equivalent values , equivalent rates and dates.
- Manipulate the exponential and logarithmic functions with proficiency.
- Calculate and manipulate quantities associated with problems on annuities (all types), perpetuities, debts, capitalization and depreciation.
Indicative Content
PAPER 1:
The aim of this paper material is to demonstrate the role of probability theory in modelling random phenomena and in statistical decision making. It begins by defining probability, sample spaces and events and some basic probability formulae. Discussion progresses onto conditional probability, independence and Bayes formula. Some counting techniques are demonstrated and those techniques put into practice in calculating probabilities. The distinction between discrete and continuous random variables is discussed along with definitions of probability distribution and expectation and variance of random variables. It explores some common discrete random variables and their probability distributions; hypergeometric, binomial, poisson, and negative binomial distributions; some common continuous random variables and their probability distributions; uniform and normal distributions. Students are provided with a brief introduction to statistical inference, sampling, and the distribution of the sample mean when sampling from a normal distribution. Properties of the Central Limit Theorem are demonstrated with applications including normal approximations to binomial distributions.
PAPER II:
This paper material begins by reviewing the normal distribution and calculations of probabilities involving normally distributed random variables and means of large random samples from virtually any population. It demonstrates methods of data summarisation and presentation, including numerical measures of location and spread for both ungrouped and grouped data, and graphical methods including histograms, stem-and-leaf and box plots. The paper material discusses statistical inference, demonstrating the explanation of statistics through practical examples of its applications, concepts of point and interval estimation, concepts in hypothesis testing including Type I and Type II errors and power, confidence intervals and hypothesis tests. This paper material aims to obtain inferences for a single population mean, a single population proportion, the difference between two population means, a single population variance and the ratio of two population variances. This paper material covers the analysis of enumerative data, including chi-squared goodness-of-fit and contingency table tests, correlation and linear regression analysis, including least squares estimation of the parameters of the simple linear regression model, inferences about these parameters, and prediction.
PAPER III:
To familiarize students with the concepts, terminology and computations involved in the mathematics of finance.
Module Resources
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