Give examples of empirical phenomena for which stochastic processes provide suitable mathematical models.

Provide an introduction to the methods of probabilistic model building for ``dynamic" events occurring over time.

Familiarise students with an important area of probability modelling.

(LO1) Apply some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes.

(LO2) Master the notions of transition matrix, equilibrium distribution, limiting behaviour etc. of a Markov chain.

(LO3) Perform calculations using special properties of the simple finite state discrete time Markov chain and Poisson processes.

(LO4) Formulate appropriate situations as probability models: random processes.

(LO5) Understand the theory underpinning simple dynamical systems.

(LO6) Select, apply and interpret results of probability techniques for a range of different problems.

(S1) Numeracy through manipulation and interpretation of datasets.

(S2) Communication through presentation of written work and preparation of diagrams

(S3) Problem solving.

(S4) Time management in the completion of the practicals and the submission of assessed work.

Material is presented during lectures (3 hours per week). Tutorials (1 hour per week) are used for consolidation and practice, and for help with individual questions.

(1) Introduction and preliminaries: Sample space, random variables, distribution functions. Conditional probabilities and expectations: definitions and properties; computing expectation by conditioning (discrete and continuous cases), computing probability by conditioning.

(2) Random walks: symmetric and asymmetric RWs, random walk with absorbing boundary: Gambler's ruin.

(3) Discrete time Markov chains: definition and examples, transition probabilities and matrices. Examples: weather model etc.

(4) Higher order transition probabilities, Chapman Kolmogorov equations.

(5) Communication of states, periodicity, recurrence and transience.

(6) Asymptotic behaviour of Markov chains, limiting and stationary distributions. Absorbing probability.

(7) Exponential distribution, memoryless property, first failure, minimum of exponential random variables.

(8) Poisson processes: definitions and examples. Interarrival time and waiting time distri butions, superposition.

(9) Poisson-like processes: compound Poisson process, non-stationary Poisson process, a selection of examples including short term insurances models.