Discrete-time stochastic processes

How do we test whether this eect is signicant if we

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Unformatted text preview: ceptual problem that was posed in Section 1.1. Suppose we have a probability model of some real-world experiment involving randomness in the sense expressed there. When the real-world experiment being modeled is performed, there is an outcome, which presumably is one of the outcomes of the probability model, but there is no observable probability. It appears to be intuitively natural, for experiments that can be carried out repeatedly under essentially the same conditions, to associate the probability of a given event with 1.5. RELATION OF PROBABILITY MODELS TO THE REAL WORLD 43 the relative frequency of that event over many repetitions. We now have the background to understand this approach. We first look at relative frequencies within the probability model, and then within the real world. 1.5.1 Relative frequencies in a probability model We have seen that for any probability model, an extended probability model exists for n IID idealized experiments of the original model. For any event A in the original model, the indicator function IA is a random variable, and the relative frequency of A over n IID experiments is the sample average of n IID rv’s with the distribution of IA . From the weak law of large numbers, this relative frequency converges in probability to E [IA ] = Pr {A}. By taking the limit n → 1, the strong law of large numbers says that the relative frequency of A converges with probability 1 to Pr {A}. In plain English, this says that for large n, the relative frequency of an event (in the nrepetition IID model) is essentially the same as the probability of that event. The word essential ly is carrying a great deal of hidden baggage. For the weak law, for any ≤, δ > 0, the relative frequency is within some ≤ of Pr {A} with a confidence level 1 − δ whenever n is sufficiently large. For the strong law, the ≤ and δ are avoided, but only by looking directly at the limit n → 1. 1.5.2 Relative frequencies in the real world In trying to sort out if and when the laws of large numbers have much to do with real-world experiments, we should ignore the mathematical details for the moment and agree that for large n, the relative frequency of an event A over n IID trials of an idealized experiment is essentially Pr {A}. We can certainly visualize a real-world experiment that has the same set of possible outcomes as the idealized experiment and we can visualize evaluating the relative frequency of A over n repetitions with large n. If that real-world relative frequency is essentially equal to Pr {A}, and this is true for the various events A of greatest interest, then it is reasonable to hypothesize that the idealized experiment is a reasonable model for the real-world experiment, at least so far as those given events of interest are concerned. One problem with this comparison of relative frequencies is that we have carefully specified a model for n IID repetitions of the idealized experiment, but have said nothing about how the real-world experiments are repeated. The IID idealized experiments specify that the conditional probability of A at one trial is the same no matter what the results of the other trials are. Intuitively, we would then try to isolate the n real-world trials so they don’t affect each other, but this is a little vague. The following examples help explain this problem and several others in comparing idealized and real-world relative frequenices. Example 1.5.1. Coin tossing: Tossing coins is widely used as a way to choose the first player in other games, and is also sometimes used as a primitive form of gambling. Its importance, however, and the reason for its frequent use, is its simplicity. When tossing a coin, we would argue from the symmetry between the two sides of the coin that each should be equally probable (since any procedure for evaluating the probability of one side 44 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY should apply equally to the other). Thus since H and T are the only outcomes (the remote possibility of the coin balancing on its edge is omitted from the model), the reasonable and universally accepted model for coin tossing is that H and T each have probability 1/2. On the other hand, the two sides of a coin are embossed in different ways, so that the mass is not uniformly distributed. Also the two sides do not behave in quite the same way when bouncing off a surface. Each denomination of each currency behaves slightly differently in this respect. Thus, not only do coins violate symmetry in small ways, but different coins violate it in different ways. How do we test whether this effect is significant? If we assume for the moment that successive tosses of the coin are well-modeled by the idealized experiment of n IID trials, we can essentially find the probability of H for a particular coin as the relative frequency of H in a sufficiently large number of independent tosses of that coin. This gives us slightly different relative frequencies for different coins, and thus slightly different probability models for different coins. If we want a generic model, we might randomly choose coi...
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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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