Discrete-time stochastic processes

The importance of the theory grew rapidly

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Unformatted text preview: es of chance. The importance of the theory grew rapidly, particularly in the 20th century, and it now plays a central role in risk assessment, statistics, data networks, operations research, information theory, control theory, theoretical computer science, quantum theory, game theory, neurophysiology, and many other fields. The core concept in probability theory is that of a probability model. Given the extent of the theory, both in mathematics and in applications, the simplicity of probability models is surprising. The first component of a probability model is a sample space, which is a set whose elements are called outcomes or sample points. Probability models are particularly simple in the special case where the sample space is finite,2 and we consider only this case in the remainder of this section. The second component of a probability model is a class of events, which can be considered for now simply as the class of all subsets of the sample space. The third component is a probability measure, which can be regarded here as the assignment of a non-negative number to each outcome, with the restriction that these numbers must sum to one over the sample space. The probability of an event is the sum of the probabilities of the outcomes comprising that event. These probability models play a dual role. In the first, the many known results about various classes of models, and the many known relationships between models, constitute the essence of probability theory. Thus one often studies a model not because of any relationship to the real world, but simply because the model provides an example or a building block useful for the theory and thus ultimately for other models. In the other role, when probability theory is applied to some game, experiment, or some other situation involving randomness, a probability model is used to represent a trial of the experiment (in what follows, we refer to all of these random situations as experiments). For example, the standard probability model for rolling a die uses {1, 2, 3, 4, 5, 6} as the sample space, with each possible outcome having probability 1/6. An odd result, i.e., the subset {1, 3, 5}, is an example of an event in this sample space, and this event has probability 1/2. The correspondence between model and actual experiment seems straightforward here. The set of outcomes is the same and, given the symmetry between faces of the die, the choice 1 It would be appealing to show how probability theory evolved from real-world situations involving randomness, but we shall find that it is difficult enough to find good models of real-world situations even using all the insights and results of probability theory as a starting basis. 2 A number of mathematical issues arise with infinite sample spaces, as discussed in the following section. 1.1. PROBABILITY MODELS 3 of equal probabilities seems natural. On closer inspection, there is the following important difference. The model corresponds to a single roll of a die, with a probability defined for each possible outcome. In a real-world single roll of a die, an outcome k from 1 to 6 occurs, but there is no observable probability of k. For repeated rolls, however, the real-world relative frequency of k, i.e., the fraction of rolls for which the output is k, can be compared with the sample value of the relative frequency of k in a model for repeated rolls. The sample values of the relative frequency of k in the model resemble the probability of k in a way to be explained later. It is this relationship through relative frequencies that helps overcome the non-observable nature of probabilities in the real world. 1.1.1 The sample space of a probability model An outcome or sample point in a probability model corresponds to the complete result of a trial of the experiment being modeled. For example, a game of cards is often appropriately modeled by the arrangement of cards within a shuffled 52 card deck, thus giving rise to a set of 52! outcomes (incredibly detailed, but trivially simple in structure), even though the entire deck might not be played in one trial of the game. A poker hand with 4 aces is an event rather than an outcome in this model, since many arrangements of the cards can give rise to 4 aces in a given hand. The possible outcomes in a probability model (and in the experiment being modeled) are mutually exclusive and collectively constitute the entire sample space (space of possible results). An outcome is often called a finest grain result of the model in the sense that an outcome ω contains no subsets other than the empty set φ and the singleton subset {ω }. Thus events typically give only partial information about the result of the experiment, whereas an outcome fully specifies the result. In choosing the sample space for a probability model of an experiment, we often omit details that appear irrelevant for the purpose at hand. Thus in modeling the set of outcomes for a coin toss as {H, T }, we usually ignore the type of coin, the ini...
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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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