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Chapter 1
RS Chapter 1 Random Variables6/14/20191 Chapter 1 Probability Theory: IntroductionBasic Probability General In a probability space ( , , P), the set is the set of all possible outcomesof a probability experiment . Mathematically, is just a set, with elements . It is called the sample space. An eventis the answer to a Yes/No question. Equivalently, an event is a subset of the probability space: A . Think of A as the set of outcomes where the answer is Yes , and Acis the complementary set where the answer is No . A -algebra is a mathematical model of a state of partial knowledge about the outcome. Informally, if is a -algebra and A , we say that A if we know whether A or Chapter 1 Random Variables6/14/20192 Definitions AlgebraDefinitions: Semiring(of sets)A collection of sets Fis called a semiringif it satisfies: F. If A, B F, then A B F. If A, B F, then there exists a collection of sets C1, C2, .., Cn F, such that A \ B = .(A\B all elements of A not in B)Definitions: AlgebraA collection of sets F is called an algebraif it satisfies: F.
All σ-algebras are algebras, and all algebras are semi-rings. Thus, if we require a set to be a semiring, it is sufficient to show instead that it is a σ-algebra or algebra. • Sigma algebras can be generated from arbitrary sets. This will be useful in developing the probability space.
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