Introduction and Review of Background Material Summary: Chapter 1 Introduction and Review of Background Material This course is an introduction to stochastic processes, with an added focus on compu- tational techniques and applications arising from biology. A stochastic process, X(t) or Xt, is a collection of random variables indexed by time, t. Most often, the time parameter t will be a subset of the nonnegative integers {0, 1, 2, . . . }, in which case it will often be denoted by n, or a subset of [0, ), the nonnegative real numbers. When time is indexed by the nonnegative integers, we say the process is discrete time, whereas when time is indexed by the nonnegative reals, we say the process is continuous time. The process Xt will take values in a state space, which can itself be either discrete (finite or countably infinite) or continuous (for example, the real line or Rd ). The main mathematical models of study in these notes are discrete and continuous time Markov chains, branching processes, point processes, and diffusion processes (those incorporating Brownian motion). Examples of how each such model arises in the biosciences will be provided throughout. We will also spend time focusing on different simulation and computational methods for the different mathematical models. The software package of choice will be Matlab, though some students may Collections: Mathematics