Discrete-time stochastic processes

More precisely we assume that for each of the

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Unformatted text preview: omplementation and countable unions and intersections. An understanding of how to construct this extension in general requires measure theory but is not necessary for our purposes. The simplest and most natural way of creating a probability measure for this extended sample space and class of events is through the assumption that the repetitions are statistically independent of each other. More precisely, we assume that for each of the extended events (A1 , A2 , . . . , An ) contained in ≠×n , we have Pr {(A1 , A2 , . . . , An )} = Yn i=1 Pr {Ai } , (1.14) where Pr {Ai } is the probability of event Ai in the original model. Note that since ≠ can be substituted for any Ai in this formula, the subset condition of (1.11) is automatically satisfied. In other words, for any probability model, there is an extended independent nrepetition model for which the events in each repetition are independent of those in the other repetitions. In what follows, we refer to this as the probability model for n independent identically distributed (IID) experiments. The niceties of how this model for n IID experiments is created depend on measure theory, but we can rely on the fact that such a model can be created, and that the events in each repetition are independent of those in the other repetitions. What we have done here is very important conceptually. A probability model for an experiment does not say anything directly about repeated experiments. However, questions about independent repeated experiments can be handled directly within this n IID experiment model. This can also be extended to a countable number of IID experiments. 1.3. PROBABILITY REVIEW 1.3.3 11 Random variables The outcome of a probabilistic experiment often specifies a collection of numerical values such as temperatures, voltages, numbers of arrivals or departures in various intervals, etc. Each such numerical value varies, depending on the particular outcome of the experiment, and thus can be viewed as a mapping from the set ≠ of sample points to the set R of real numbers. Note that R does not include ±1, and in those rare instances where ±1 should be included, it is called the extended set of real numbers. These mappings from sample points to numerical values are called random variables. Definition 1.4. A random variable (rv) is a function X from the sample space ≠ of a probability model to the set of real numbers R. That is, each sample point ω in ≠, except perhaps for a subset11 of probability 0, is mapped into a real number denoted by X (ω ). There is an additional technical requirement that, for each real x, the set {X (ω ) ≤ x} must be an event in the probability model. As with any function, there is often some confusion between the function itself, which is called X in the definition above, and the value X (ω ) the function takes on for a sample point ω . This is particularly prevalent with random variables (rv’s) since we intuitively associate a rv with its sample value when an experiment is performed. We try to control that confusion here by using X , X (ω ), and x respectively, to refer to the rv, the sample value taken for a given sample point ω , and a generic sample value. Definition 1.5. The distribution function12 FX (x) of a random variable (rv) X is a function, R → R, defined by FX (x) = Pr {ω ∈ ≠ | X (ω ) ≤ x}. The argument ω is usual ly omitted for brevity, so FX (x) = Pr {X ≤ x}. Note that x is the argument of FX (x), whereas the subscript X denotes the particular rv under consideration. As illustrated in Figure 1.1, the distribution function FX (x) is nondecreasing with x and must satisfy the limits limx→−1 FX (x) = 0 and limx→1 FX (x) = 1 (see Exercise 1.3). The concept of a rv is often extended to complex random variables (rv’s) and vector rv’s. A complex random variable is a mapping from the sample space to the set of finite complex numbers, and a vector random variable (rv) is a mapping from the sample space to the finite vectors in some finite dimensional vector space. Another extension is that of defective rvs. X is defective if there is an event of positive probability for which the mapping is either undefined or defined to be either +1 or −1. When we refer to random variables in this text (without any modifier such as complex, vector, or defective), we explicitly restrict attention to the original definition, i.e., a function from ≠ to R. If X has only a finite or countable number of possible sample values, say x1 , x2 , . . . , the probability Pr {X = xi } of each sample value xi is called the probability mass function 11 We will find many situations, particularly when taking limits, when rv’s are undefined for some set of sample points of probability zero. These are still called rv’s (rather than defective rv’s ) and the zero probability subsets can usually be ignored. 12 Some people refer to the distribution function as the cumulative distribution function. 12 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY 1 FX (x) 0 Figure 1.1: Example of a distribution function for a rv that is neith...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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