Discrete-time stochastic processes

# What the word approaches means here is both tricky

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Unformatted text preview: lly, most of the random processes to be studied in this text are deﬁned as various ways of combining simpler idealized experiments. What is really happening as we look at modeling increasingly sophisticated systems and studying increasingly sophisticated models is that we are developing mathematical results for simple idealized models and relating those results to real-world results (such as relating idealized statistically independent trials to real-world independent trials). The association of relative frequencies to probabilities forms the basis for this, but is usually exercised only in the simplest cases. The way one selects probability models of real-world experiments in practice is to use scientiﬁc knowledge and experience, plus simple experiments, to choose a reasonable model. The results from the model (such as the law of large numbers) are then used both to hypothesize results about the real-world experiment and to provisionally reject the model when further experiments show it to be highly questionable. Although the results about the model are mathematically precise, the corresponding results about the real-world are at best insightful hypotheses whose most important aspects must be validated in practice. 1.5.5 Sub jective probability There are many useful applications of probability theory to situations other than repeated trials of a given experiment. When designing a new system in which randomness (of the type used in probability models) is hypothesized to be signiﬁcantly involved, one would like to analyze the system before actually building it. In such cases, the real-world system does not exist, so indirect means must be used to construct a probability model. Often some sources of randomness, such as noise, can be modeled in the absence of the system. Often similar systems or simulation can be used to help understand the system and help in formulating appropriate probability models. However, the choice of probabilities is to a certain extent sub jective. In other situations, as illustrated in the examples above, there are repeated trials of similar experiments, but a probability model would have to choose one probability assignment for a model rather than a range of assignments, each appropriate for diﬀerent types of experiments. Here again the choice of probabilities for a model is somewhat sub jective. Another type of situation, of which a canonic example is risk analysis for nuclear reactors, deals with a large number of very unlikely outcomes, each catastrophic in nature. Experimentation clearly cannot be used to establish probabilities, and it is not clear that probabilities have any real meaning here. It can be helpful, however, to choose a probability model on the basis of sub jective beliefs which can be used as a basis for reasoning about the problem. When handled well, this can at least make the sub jective biases clear, leading to a more rational approach to the problem. When handled poorly, it can hide the arbitrary 48 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY nature of possibly poor decisions. We will not discuss the various, often ingenious methods to choose sub jective probabilities. The reason is that sub jective beliefs should be based on intensive and long term exposure to the particular problem involved; discussing these problems in abstract probability terms weakens this link. We will focus instead on the analysis of idealized models. These can be used to provide insights for sub jective models, and more reﬁned and precise results for ob jective models. 1.6 Summary This chapter started with an introduction into the correspondence between probability theory and real-world experiments involving randomness. While almost all work in probability theory works with established probability models, it is important to think through what these probabilities mean in the real world, and elementary sub jects rarely address these questions seriously. The next section discussed the axioms of probability theory, along with some insights about why these particular axioms were chosen. This was followed by a review of conditional probabilities, statistical independence, rv’s, stochastic processes, and expectations. The emphasis was on understanding the underlying structure of the ﬁeld rather than reviewing details and problem solving techniques. This was followed by a fairly extensive treatment of the laws of large numbers. This involved a fair amount of abstraction, combined with mathematical analysis. The central idea is that the sample average of n IID rv’s approaches the mean with increasing n. As a special case, the relative frequency of an event A approaches Pr {A}. What the word approaches means here is both tricky and vital in understanding probability theory. The strong law of large numbers requires mathematical maturity, and might be postponed to Chapter 3 where it is ﬁrst used. The ﬁnal section came back to the fundamental problem of understanding the relation between probability theory and randomn...
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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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