General description
This module aims to develop common probability models, applicable to a variety of situations and to illustrate their use in statistical inference. It also includes an introduction to the theory of estimation.

Aims
To introduce the subject of Statistics to mathematics students.

Learning outcomes
On completion of this module, a student should be able to:

• describe the notion of covariance;
• calculate means and variances of linear combinations of random variables;
• identify a probability distribution appropriate to a given situation;
• describe modelling in terms of Bernoulli trials and of random events;
• manipulate distributions to obtain moments and to sketch curves;
• assess a given value in relation to the scale of a given probability distribution;
• estimate means and proportions from data;
• explain the use of a statistical test;
• construct and carry out simple tests.

Syllabus
1. THE INFERENCE PROBLEM: The difference between probability and statistical inference. The idea of likelihood. Assessing 'typical' values from a distribution. The idea of a statistics. Estimates and estimators. Accuracy and precision. Bias, sampling, variance and mean squared error. Comparison of estimators.
2. PROBABILISTIC (STOCHASTIC) MODELLING (INCLUDING EXAMPLES OF INFERENCE): Bernoulli trials and distributions based on them (Geometric, Binomial). Opinion polls. The ideas of covariance and correlation. Variances of linear combinations of random variables. Modelling random events. The Poisson and exponential distributions. Normality and the Central Limit Theorem. The Weak Law of Large Numbers.
3. INFERENCE: Sampling mean, sampling variance and standard deviation of a sample total and a sample average. Statistical testing. Tail areas. p-values.