1: Introduction and Review of Probability
- Page ID
- 44600
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- 1.1: Probability Models
- Probability theory is a central field of mathematics, widely applicable to scientific, technological, and human situations involving uncertainty. The most obvious applications are to situations, such as games of chance, in which repeated trials of essentially the same procedure lead to differing outcomes.
- 1.2: The Axioms of Probability Theory
- As the applications of probability theory became increasingly varied and complex during the 20th century, the need arose to put the theory on a firm mathematical footing. This was accomplished by an axiomatization of the theory, successfully carried out by the great Russian mathematician A. N. Kolmogorov in 1932.
- 1.4: Basic Inequalities
- Inequalities play a particularly fundamental role in probability, partly because many of the models we study are too complex to find exact answers, and partly because many of the most useful theorems establish limiting rather than exact results. In this section, we study three related inequalities, the Markov, Chebyshev, and Chernoff bounds.
- 1.6: Relation of Probability Models to the Real World
- Whenever experienced and competent engineers or scientists construct a probability model to represent aspects of some system that either exists or is being designed for some application, they must acquire a deep knowledge of the system and its surrounding circumstances, and concurrently consider various types of probability models used in probabilistic analysis of the same or similar systems.
- 1.7: Summary
- This chapter started with an introduction into the correspondence between probability theory and real-world experiments involving randomness. While almost all work in probability theory works with established probability models, it is important to think through what these probabilities mean in the real world, and elementary subjects rarely address these questions seriously.