56 CHAPTER 1. BACKGROUND IDEAS
Key Concepts
- A sequence or interval of random outcomes, that is to say, a string
of random outcomes dependent on time as well as the randomness
is called astochastic process. Because of the inclusion of a time
variable, the rich range of random outcome distributions is multiplied
to an almost bewildering variety of stochastic processes. Nevertheless,
the most commonly studied types of random processes do have a family
tree of relationships. - Stochastic processes are functions of two variables, the time index and
the sample point. As a consequence, there are several ways to repre-
sent the stochastic process. The simplest is to look at the stochastic
process at a fixed value of time. The result is a random variable with
a probability distribution, just as studied in elementary probability. - Another way to look at a stochastic process is to consider the stochas-
tic process as a function of the sample pointω. For eachωthere is an
associated function of timeX(t). This means that one can look at a
stochastic process as a mapping from the sample space Ω to a set of
functions. In this interpretation, stochastic processes are a generaliza-
tion from the random variables of elementary probability theory.
Vocabulary
- A sequence or interval of random outcomes, that is, random outcomes
dependent on time is called astochastic process. - LetJ be a subset of the non-negative real numbers. Let Ω be a set,
usually called thesample spaceorprobability space. An element
ωof Ω is called asample pointorsample path. LetSbe a set of
values, often the real numbers, called thestate space. Astochastic
processis a functionX: (J,Ω)→S, that is a function of both time
and the sample point to the state space. - The particular stochastic process usually called a simple random
walkTn gives the position in the integers after taking a step to the
right for a head, and a step to the left for a tail.