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) Numbered Display Equation

where X[k] (with ) is the binomial counting sequence in (6.160). The ...

Get Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.