November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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6.12 RANDOM PROCESSES
Next, we consider some random processes that evolve continuously over time. As was the case with random sequences, the time-indexed random variables can have discrete or continuous outcomes. We begin with a sequence of independent random variables having a Poisson distribution, and then allow the intervals between time instants to approach zero in order to generate a continuous-time random process that can have a discrete change in amplitude at any time instant. (Note that example plots of the various random processes are actually discrete-time realizations because, obviously, MATLAB can only plot a finite number of outcomes for a process.)
6.12.1 Poisson Counting Process
The outcomes of the Poisson random variable are discrete (countably infinite), given by all nonnegative integers 
. Define the time interval 
where 
is continuous time and 
. Consider the random process defined as follows:
(6.172) 
where X[k] (with ) is the binomial counting sequence in (6.160). The ...
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