University of Bristol - MATH10013: Probability and Statistics (Solutions)

This unit, MATH10013: Probability and Statistics - introduces student to basic ideas and methods in the areas of Probability and Statistics. It aims to develop the concepts of random variables, expectations and variances and look at some simple applications of these ideas and methods. It will also introduce the role of statistics in contemporary applications and aims to develop an elementary understanding of, and fluency in, the statistical paradigm of data collection, exploration, modelling and inference.

At the end of the unit, you should be able to:

• understand the basic framework of modern probability theory, including random variables, expectations, probability mass/ density functions, conditioning, and independence;
• define the following random variables: Bernoulli, Binomial, Geometric, Poisson, Uniform, Exponential, Gamma, Normal/ Gaussian. Recall and illustrate features of these distributions;
• define jointly distributed random variables, joint probability mass functions;
• understand how to analyse sums of independent random variables, including using moment generating functions and conditioning;
• formulate formal probability models from informal descriptions, and formulate simple statistical models as appropriate to particular applications;
• use exploratory techniques to identify simple relationships in data;
• understand the principles of parametric modelling, and be able to derive parameter estimates for simple models using method-of-moments and maximum likelihood;
• derive the simple linear regression model and implement it in appropriate situations;
• understand estimators and sample variability, confidence intervals, and hypothesis tests, using both closed-form expressions for simple models, and simulation methods; and, 
• use the statistical software system R to support each of the above tasks.