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lecture15 - ECE 5510 Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 15 Today: (1) Random Process Intro (2) Binomial R.P. (3) Pois- son R.P. 1 Random Processes This starts into Chapter 10, ‘Stochastic’ Processes. As Y&G says, ”The word stochastic means random.” So I prefer ‘Random Pro- cesses’. We’ve covered Random Vectors, which have many random variables. So what’s new? • Before we had a few random variables, X 1 ,X 2 ,X 3 . Now we have possibly infinitely many: X 1 ,X 2 ,... . • In addition, we may not be taking samples - we may have a continuously changing random variable, indexed by time t . We’ll denote this as X ( t ). Def’n: Random Process A random process X ( t ) consists of an experiment with a probability measure P [ · ], a sample space S , and a function that assigns a time (or space) function x ( t,s ) to each outcome s in the sample space. Recall that we used S to denote the event space, and every s ∈ S is a possible ‘way’ that the outcome could occur. 1.1 Continuous and Discrete-Time Types of Random Processes: A random process (R.P.) can be either 1. Discrete-time : Samples are taken at particular time in- stants, for example, t n = nT where n is an integer and T is the sampling period. In this case, rather than referring to X ( t n ), we abbreviate it as X n . (This matches exactly our previous notation.) In this case, we also call it a random se- quence . 2. Continuous-time : Uncountably-infinite values exist, for ex- ample, for t ∈ (0 , ∞ ). ECE 5510 Fall 2009 2 Types of Random Processes: A random process (R.P.) can be still be 1. Discrete-valued : The sample space S X is countable. That is, each value in the R.P. is a discrete r.v. (For example, our R.P. can only take integer values, or we allow a finite number of decimal places.) 2. Continuous-valued : The sample space S X is uncountably infinite. Draw a example plot here of each of the following: Discrete-Time Continuous-Time Discrete-Valued Continuous-Valued 1.2 Examples For each of these examples, say whether this is continuous/discrete time, and continuous/discrete valued (there may be multiple ‘right’ answers): • Stock Values : Today, we invest $1000 in one stock. Let X ( t...
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lecture15 - ECE 5510 Random Processes Lecture Notes Fall...

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