21-01-2013, 12:03 PM
Introduction to Stochastic Processes
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Stochastic Processes
A random variable is a mapping function which assigns outcomes of a random
experiment to real numbers (see Fig. 1). Occurrence of the outcome follows
certain probability distribution. Therefore, a random variable is completely
characterized by its probability density function (PDF).
Figure1. A random variable is a function mapping outcomes of a random
experiment to real numbers.
Characterizations of a Stochastic Processes
First-order densities of a random process
A stochastic process is defined to be completely or totally characterized if the
joint densities for the random variables ( ), ( ), ( ) 1 2 n X t X t X t are known for all
times n t ,t , ,t 1 2 and all n. In general, a complete characterization is practically
impossible, except in rare cases. As a result, it is desirable to define and work with
various partial characterizations. Depending on the objectives of applications, a
partial characterization often suffices to ensure the desired outputs.