 
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
