Students must normally have A-level Mathematics, Grade A, or equivalent, in order to undertake this unit.
Aims & Learning Objectives:
Aims: To provide a solid foundation in discrete probability theory that will facilitate further study in probability and statistics.
Students should be able to: apply the axioms and basic laws of probability using proper notation and rigorous arguments; solve a variety of problems with probability, including the use of combinations and permutations and discrete probability distributions; perform common expectation calculations; calculate marginal and conditional distributions of bivariate discrete random variables; calculate and make use of some simple probability generating functions.
Sample space, events as sets, unions and intersections. Axioms and laws of probability. Equally likely events. Combinations and permutations. Conditional probability. Partition Theorem. Bayes' Theorem. Independence of events. Bernoulli trials. Discrete random variables (RVs). Probability mass function (PMF). Bernoulli, Geometric, Binomial and Poisson Distributions. Poisson limit of Binomial distribution. Hypergeometric Distribution. Negative binomial distribution. Joint and marginal distributions. Conditional distributions. Independence of RVs. Distribution of a sum of discrete RVs. Expectation of discrete RVs. Means. Expectation of a function. Moments. Properties of expectation. Expectation of independent products. Variance and its properties. Standard deviation. Covariance. Variance of a sum of RVs, including independent case. Correlation. Conditional expectations. Probability generating functions (PGFs).