Introduction to Probability Theory
In this chapter we summarize the most important notions and facts of probability theory that are necessary for an elaboration of our topic. In the present summary, we will apply the more specific mathematical concepts and facts – mainly measure theory and
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Introduction to Probability Theory
1.1 Summary of Basic Notions of Probability Theory In this chapter we summarize the most important notions and facts of probability theory that are necessary for an elaboration of our topic. In the present summary, we will apply the more specific mathematical concepts and facts – mainly measure theory and analysis – only to the necessary extent while, however, maintaining mathematical precision. Random Event We consider experiments whose outcomes are uncertain, where the totality of the circumstances that are or can be considered does not determine the outcome of the experiment. A set consisting of all possible outcomes is called a sample space. We define random events (events for short) as certain sets of outcomes (subsets of the sample space). It is assumed that the set of events is closed under countable set operations, and we assign probability to events only; they characterize the quantitative measure of the degree of uncertainty. Henceforth countable means finite or countably infinite. Denote the sample space by D f!g. If is countable, then the space is called discrete. In a mathematical approach, events can be defined as subsets A of the possible outcomes having the properties (-algebra properties) defined subsequently. A given event A occurs in the course of an experiment if the outcome of the experiment belongs to the given event, that is, if an outcome ! 2 A exists. An event is called simple if it contains only one outcome !. It is always assumed that the whole set and the empty set ¿ are events that are called a certain event and an impossible event, respectively. Operation with Events; Notion of -Algebra Let A and B be two events. The union A [ B of A and B is defined as an event consisting of all elements ! 2 belonging to either event A or B, i.e., A [ B D f! W ! 2 A or ! 2 Bg. The intersection (product) A\B .AB/ of events A and B is defined as an event consisting of all elements ! 2 belonging to both A and B, i.e., L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, DOI 10.1007/978-1-4614-5317-8 1, © Springer Science+Business Media, LLC 2013
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1 Introduction to Probability Theory
A \ B D f! W ! 2 A and ! 2 Bg: The difference AnB, which is not a symmetric operation, is defined as the set of all elements ! 2 belonging to event A but not to event B, i.e., AnB D f! W ! 2 A and ! … Bg: A complementary event A of A is defined as a set of all elements ! 2 that does not belong to A, i.e., A D nA: If A \ B D ˛, then sets A and B are said to be disjoint or mutually exclusive. Note that the operations [ and \ satisfy the associative, commutative, and distributive properties .A [ B/ [ C D A [ .B [ C /; A [ B D B [ A;
and .A \ B/ \ C D A \ .B \ C /; and A \ B D B \ A;
A \ .B [ C / D .A \ B/ [ .A \ C /; and A [ .B \ C / D .A [ B/ \ .A [ C /: DeMorgan identities are valid also for the operations union, intersection, and complementarity of events as follows: A [ B D A \ B;
A \ B D A [ B:
With the use of the p
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