Discrete-time stochastic processes

Example 121 suppose we want to model the phase of a

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Unformatted text preview: to each sample point, often fails with infinite sample spaces. Example 1.2.1. Suppose we want to model the phase of a sine wave, where the phase is reasonably viewed as being uniformly distributed between 0 and 2π . If this phase is the only quantity of interest, it is reasonable to choose a sample space consisting of the set of real numbers between 0 and 2π . There are an uncountably5 infinite number of possible phases between 0 and 2π , and with any reasonable interpretation of uniform distribution, one must conclude that each sample point has probability zero. Thus, the simple approach of the last section leads us to conclude that any event with a finite or countably infinite set of sample points should have probability zero, but that there is no way to find the probability of, say, the interval (0, π ). For this example, the appropriate view, as developed in all elementary probability texts, is to assign a probability density 21 to the phase. The probability of any given event of π interest can usually be found by integrating the density over the event, whereas it generally can not be found from knowing only that each sample point has 0 probability. Useful as densities are, however, they do not lead to a general approach over arbitrary sample spaces.6 Example 1.2.2. Consider an infinite sequence of coin tosses. The probability of any given n-tuple of individual outcomes can be taken to be 2−n , so in the limit n = 1, the probability of any given sequence is 0. There are 2n equiprobable outcomes for n tosses and an infinite number of outcomes for n = 1. Again, expressing the probability of events involving infinitely many tosses as a sum of individual sample point probabilities does not work. The obvious approach is to evaluate the probability of an event as an appropriate limit from finite length sequences, but we find situations in which this doesn’t work. Although appropriate rules can be generated for simple examples such as those above, there is a need for a consistent and general approach. In such an approach, rather than assigning probabilities to sample points, which are then used to assign probabilities to events, probabilities must be associated directly with events. The axioms to follow establish consistency requirements between the probabilities of different events. The axioms, and 5 A set is uncountably infinite if it is infinite but its members cannot be put into one-to-one correspondence with the positive integers. For example the set of real numbers over some interval such as (0, 2π ) is uncountably infinite. The Wikipedia article on countable sets provides a friendly introduction to the concepts of countability and uncountability. 6 It is possible to avoid the consideration of infinite sample spaces here by quantizing the possible phases. This is analogous to avoiding calculus by working only with discrete functions. Both usually result in both artificiality and added complexity. 6 CHAPTER 1. INTRODUCTION AND REVIEW OF PROBABILITY the corollaries derived from them, are consistent with one’s intuition, and, for finite sample spaces, are consistent with our earlier approach. Dealing with the countable unions of events in the axioms will be unfamiliar to some students, but will soon become both familiar and consistent with intuition. The strange part of the axioms comes from the fact that defining the class of events as the set of al l subsets of the sample space is usually inappropriate when the sample space is uncountably infinite. What is needed is a class of events that is large enough that we can almost forget that some very strange subsets are excluded. This is accomplished by having two simple sets of axioms, one defining the class of events,7 and the other defining the relations between the probabilities assigned to these events. In this theory, all events have probabilities, but those truly weird subsets that are not events do not have probabilities. This will be discussed more after giving the axioms for events. The axioms for events use the standardS notation of set theory. That is, for a set ≠, the union of two subsets A1 and A2 , denoted A1 A2 , isT subset containing all points in either A1 the or A2 . The intersection, denoted A1 A2 , or A1 A2 is the set of points in both A1 and A2 . A sequence of events is a class of events that can be put into one-to-one correspondence with the positive integers, i.e., A1 , A2 , . . . , ad infinitum. The union of a sequence of events is called a countable union, meaning the union includes one event for each positive integer n. 1.2.1 Axioms for the class of events for a sample space ≠: 1. ≠ is an event. 2. For every sequence of events A1 , A2 , . . . , the union S1 n=1 An is an event. 3. For every event A, the complement of A, denoted Ac is an event. There are a number of important corollaries of these axioms. The first is that since the empty set φ is the complement of ≠, it is also an event. The empty set does not correspond to our intuition about events, but the theory would be extremely awkwar...
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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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