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Fundamentals of Stochastic Networks by Oliver C. Ibe

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2

OVERVIEW OF STOCHASTIC PROCESSES

2.1 INTRODUCTION

Stochastic processes deal with the dynamics of probability theory. The concept of stochastic processes enlarges the random variable concept to include time. Thus, instead of thinking of a random variable X that maps an event w ∈ Ω, where Ω is the sample space, to some number X(w), we think of how the random variable maps the event to different numbers at different times. This implies that instead of the number X(w) we deal with X(t, w), where tT and T is called the parameter set of the process and is usually a set of times.

Stochastic processes are widely encountered in such fields as communications, control, management science, and time series analysis. Examples of stochastic processes include the population growth, the failure of equipment, the price of a given stock over time, and the number of calls that arrive at a switchboard.

If we fix the sample point w, we obtain X(t), which is some real function of time; and for each w, we have a different function X(t). Thus, X(t, w) can be viewed as a collection of time functions, one for each sample point w. On the other hand, if we fix t, we have a function X(w) that depends only on w and thus is a random variable. Thus, a stochastic process becomes a random variable when time is fixed at some particular value. With many values of t we obtain a collection of random variables. Thus, we can define a stochastic process as a family of random variables {X(t, w)|tT, w ∈ Ω} defined ...

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