Statistics
ST112: Probability (5 ECTS)
The module 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.
Taught in Semester(s) I. Examined in Semester(s) I.
Workload: 24 hours (24 Lecture hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- 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 hypergeometric, 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
Indicative Content
General Aims:
The aim of this module is to demonstrate the role of probability theory in modelling random phenomena and in statistical decision making. The module 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. The module 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 a 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.
Module Resources
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