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Stochastic Process Definition

 

A stochastic process is defined as an indexed collection of random variables {X1 }, parameterized on time t, which are defined on a common sample space.

 

It is to be noted that the distribution of the random variable X1 may not be the same at different points in time t1 and 12. If they are the same, we refer to the random variables as identically distributed. To specify a stochastic process completely, we must also distinguish when the samples of the random variable occur in time (the embedding points). The points in time may be equally spaced or their spacing may depend upon the overall behavior of the physical system in which the stochastic process is embedded.

 

Example 1. The Poisson process described in section 2 of Chapter 8 represents a stochastic process with an infinite number of states. Assuming the process starts at time (zero) 0, the random variable X represents the occurrences (arriving of customers) between 0 and t. The states of the system at any time t are thus given by x = 0, 1, 2 . ...... .

 

Example 2. The successive roll of a pair of dice in a game is a stochastic process. Each roll has the same sample space, i. e., from 2 to 12 and each roll occurs at some point in lime. Each of the random variables is identically distributed and independent.