# M3S4 - Applied Probability

## Applied Probability

### Module aims

This course aims to give students an understanding of the basics of stochastic processes.  The theory of different kinds of processes will be described, and will be illustrated by applications in several areas. The groundwork will be laid for further deep work, especially in such areas as genetics, finance, industrial applications, and medicine.

### Module syllabus

Revision of basic ideas of probability. Important discrete and continuous probability distributions. Random processes: Bernoulli processes, point processes. Poisson processes and their properties; Superposition, thinning of Poisson processes; Non-homogeneous, compound, and doubly stochastic Poisson processes. Autocorrelation functions. Probability generating functions and how to use them. General continuous-time Markov chains: generator, forward and backward equations, holding times, stationarity, long-term behaviour, jump chain, explosion; birth, death, immigration, emigration processes. Differential and difference equations and pgfs. Finding pgfs. Embedded processes. Time to extinction. Queues. Brownian motion and its properties. Random walks. Gambler’s ruin. Branching processes and their properties. Galton-Watson model. Absorbing and reflecting barriers. Markov chains. Chapman-Kolmogorov equations. Recurrent, transient, periodic, aperiodic chains. Returning probabilities and times. Communicating classes. The basic limit theorem. Stationarity. Ergodic Theorem. Time-reversibility.