Abstract: Previous chapter Next chapter Advances in Design and Control Stochastic Processes, Estimation, and Control1. Probability Theorypp.1 - 23Chapter DOI:https://doi.org/10.1137/1.9780898718591.ch1PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt In this introductory chapter the concept of a probability space is defined. This notion plays an important role in understanding the underlying foundation that probability theory plays in the more advanced algorithms of estimation and stochastic control. This probability space will be composed of a sample space of elementary events, an algebra constructing more complex events, and a probability measure associated with the events in the algebra. By using the concept of a probability space, notions such as joint, marginal, and conditional probability and Bayes' rule are introduced. 1.1 Probability Theory as a Set of Outcomes Most of us by now possess some intuitive notion of probability. They are percentages or "odds," the chance that something will happen. The mathematical notion of probability is built around sets, or collections of things. What we intuitively understand as an event or outcome is a subset of the set of all possible outcomes. The power of set theory is that a set can range from a single element to an infinite number of elements (though infinity comes in many flavors) and that the members of a set can be absolutely anything. A probability, the number that ranges from 0 to 1, then becomes a restricted version of a function on sets called measures. A measure gives us the size of a set. A probability measure is a measure that by definition must be finite and is by convention normalized, i.e., no greater than 1. The restriction to being finite, as we will see, makes things a little tricky when the sets we work with have an infinite number of members. We will see that this approach to probability jibes with our ingrained notions and perhaps even sharpens them. In this view, probability describes the significance of one choice out of a set of many choices. If you are rolling a single die, you can easily see that it has six sides. Hence, the probability of having any one of the six faces lie upwards is one in six. Of course, there are drawbacks to our set-based probability. Previous chapter Next chapter RelatedDetails Published:2008ISBN:978-0-89871-655-9eISBN:978-0-89871-859-1 https://doi.org/10.1137/1.9780898718591Book Series Name:Advances in Design and ControlBook Code:DC17Book Pages:xiv + 383Key words:Probability theory, Stochastic processes, Estimation, Filtering theory, Stochastic optimal control
Publication Year: 2008
Publication Date: 2008-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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