60 CHAPTER 1. BACKGROUND IDEAS
up to timetby using a Geiger counter. The discrete state variable is the
counting number of clicks heard. The mathematical “Poisson process” is an
excellent model of this physical process. More generally, instead of radioac-
tive events giving a single daughter particle, we can imagine a birth event
with a random number (distributed according to some probability law) of
offspring born at random times. Then the stochastic process measures the
population in time. These are called “birth processes” and make excellent
models in population biology and the physics of cosmic rays. We can con-
tinue to generalize and imagine that each individual in the population has
a random life-span distributed according to some law, then dies. This gives
a “birth-and-death process”. In another variation, we can imagine a dis-
ease with a random number of susceptibles getting infected, in turn infecting
a random number of others, then recovering and becoming immune. The
stochastic process counts how many of each type there are at any time, an
“epidemic process”.
In another variation, we can consider customers arriving at a service
counter at random intervals with some specified distribution, often taken to
be an exponential probability distribution with parameterλ. The customers
are served one-by-one, each taking a random service time, again often taken
to be exponentially distributed. The state space is the number of customers
waiting to be served, the queue length at any time. These are called “queuing
processes”. Mathematically, many of these processes can be studied by what
are called “compound Poisson processes”.
Continuous Space Processes usually take the state space to be the real
numbers or some interval of the reals. One example is the magnitude of noise
on top of a signal, say a radio message. In practice the magnitude of the noise
can be taken to be a random variable taking values in the real numbers, and
changing in time. Then subtracting off the known signal, we would be left
with a continuous-time, continuous state-space stochastic process. In order
to mitigate the noise’s effect engineers will be interested in modeling the
characteristics of the process. To adequately model noise the probability
distribution of the random magnitude has to be specified. A simple model is
to take the distribution of values to be normally distributed, leading to the
class of “Gaussian processes” including “white noise”.
Another continuous space and continuous time stochastic process is a
model of the motion of particles suspended in a liquid or a gas. The random
thermal perturbations in a liquid are responsible for a random walk phe-
nomenon known as “Brownian motion” and also as the “Wiener process”,