Discrete-time stochastic processes

This leads for example to the ability to calculate

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Unformatted text preview: The examples above show that the problems of modeling a real-world experiment are often connected with the question of creating a model for a set of experiments that are not exactly the same and do not necessarily correspond to the notion of independent repetitions within the model. In other words, the question is not only whether the probability model is reasonable for a single experiment, but also whether the IID repetition model is appropriate for multiple copies of the real-world experiment. At least we have seen, however, that if a real-world experiment can be performed many times with a physical isolation between performances that is well modeled by the IID repetition model, then the relative frequencies of events in the real-world experiment correspond to relative frequencies in the idealized IID repetition model, which correspond to probabilities in the original model. In other words, under appropriate circumstances, the probabilities in a model become essentially observable over many repetitions. We will see later that our emphasis on IID repetitions was done for simplicity. There are other models for repetitions of a basic model, such as Markov models, that we study later. 46 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY These will also lead to relative frequencies approaching probabilities within the repetition model. Thus, for repeated real-world experiments that are well modeled by these repetition models, the real world relative frequencies approximate the probabilities in the model. 1.5.3 Statistical independence of real-world experiments We have been discussing the use of relative frequencies of an event A in a repeated realworld experiment to test Pr {A} in a probability model of that experiment. This can be done essentially successfully if the repeated trials correpond to IID trials in the idealized experiment. However, the statement about IID trials in the idealized experiment is a statement about probabilities in the extended n trial model. Thus, just as we tested Pr {A} by repeated real-world trials of a single experiment, we should be able to test Pr {A1 , . . . , An } in the n-repetition model by a much larger number of real-world repetitions taking an n-tuple at a time. To be more specific, choose two large integers, m and n, and perform the underlying realworld experiment mn times. Partition the mn trials into m runs of n trials each. For any given n-tuple A1 , . . . , An of successive events, find the relative frequency (over m trials of n tuples) of the n-tuple event A1 , . . . , An . This can then be used essentially to test the probability Pr {A1 , . . . , An } in the model for n IID trials. The individual event probabilities can also be tested, so the condition for independence can be tested. The observant reader will note that there is a tacit assumption above that successive n tuples can be modeled as independent, so it seems that we are simply replacing a big problem with a bigger problem. This is not quite true, since if the trials are dependent with some given probability model for dependent trials, then this test for independence will essentially reject the independence hypothesis for large enough n. Choosing models for real-world experiments is primarily a sub ject for statistics, and we will not pursue it further except for brief discussions when treating particular application areas. The purpose here was to treat a fundamental issue in probability theory. As stated before, probabilities are non-observables — they exist in the theory but are not directly measurable in real-world experiments. We have shown that probabilities essentially become observable in the real-world via relative frequencies over repeated trials. 1.5.4 Limitations of relative frequencies Most real-world applications that are modeled by probability models have such a large sample space that it is impractical to conduct enough trials to choose probabilities from relative frequencies. Even a shuffled deck of 52 cards would require many more than 52! ≈ 8 × 1067 trials for most of the outcomes to appear even once. Thus relative frequencies can be used to test the probability of given individual events of importance, but are usually impractical for choosing the entire model and even more impractical for choosing a model for repeated trials. Since relative frequencies give us a concrete interpretation of what probability means, however, we can now rely on other approaches, such as symmetry, for modeling. From symmetry, 1.5. RELATION OF PROBABILITY MODELS TO THE REAL WORLD 47 for example, it is clear that all 52! possible arrangements of a card deck are equiprobable after shuffling. This leads, for example, to the ability to calculate probabilities of different poker hands, etc., which are such popular exercises in elementary probability. Another valuable modeling procedure is that of constructing a probability model where the possible outcomes are independently chosen n-tuples of outcomes in a simpler model. More genera...
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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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