Aims & Learning Objectives:
Aims: To introduce probability theory for continuous random variables. To introduce statistical modelling and parameter estimation and to discuss the role of statistical computing.
Ability to solve a variety of problems and compute common quantities relating to continuous random variables. Ability to formulate, fit and assess some statistical models. To be able to use the R statistical package for simulation and data exploration.
Definition of continuous random variables (RVs), cumulative distribution functions (CDFs) and probability density functions (PDFs). Some common continuous distributions including uniform, exponential and normal. Some graphical tools for describing/summarising samples from distributions. Results for continuous RVs analogous to the discrete RV case, including mean, variance, properties of expectation, joint PDFs (including dependent and independent examples), independence (including joint distribution as a product of marginals). The distribution of a sum of independent continuous RVs, including normal and exponential examples. Statement of the central limit theorem (CLT). Transformations of RVs. Discussion of the role of simulation in statistics. Use of uniform random variables to simulate (and illustrate) some common families of discrete and continuous RVs. Sampling distributions, particularly of sample means. Point estimates and estimators. Estimators as random variables. Bias and precision of estimators. Introduction to model fitting; exploratory data analysis (EDA) and model formulation. Parameter estimation via method of moments and (simple cases of) maximum likelihood. Graphical assessment of goodness of fit. Implications of model misspecification.