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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.

RS – Chapter 1 – Random Variables 6/14/2019 5 Definition: Borel σ-algebra (Emile Borel (1871-1956), France.) The Borel σ-algebra (or, Borel field) denoted B, of the topological space (X; τ) is the σ-algebra generated by the family τof open sets. Its elements are called Borel sets.

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