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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.

neighborhoods, open sets, closed set, etc. We present the definition based on open sets. (Ω, Σ) Definition (via open sets): A topological space is an ordered pair (X, τ), where X is a set and τis a collection of subsets ofX, satisfying: 1. Ø; X ∈τ 2. τis closed under finite intersections. 3. τis closed under arbitrary unions.

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