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Chap1 - Chapter 1 General Probability Theory Denition 1.1.1...

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Chapter 1. General Probability Theory Definition 1.1.1. Let Ω be a nonempty set, and let F be a collection of subsets of Ω. We say F is a σ -algebra (or field) provided that (i) the empty set belongs to F ; (ii) whenever a set A belongs to F , its com- plement A c also belongs to F ; (iii) whenever a sequence of sets A 1 , A 2 , · · · be- longs to F , their union i =1 U i belongs to F . 1
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Definition 1.1.2. Let Ω be a nonempty set, and let F be a σ -field of subsets of Ω. A prob- ability measure P is a fucntion that, to every set A ⊂ F , assigns a number in [0 , 1], called the probability of A and write P ( A ). We require (i) P (Ω) = 1 (ii) whenever A 1 , A 2 , · · · is a sequence of dis- joint sets in F , then P ( n =1 A n ) = n =1 P ( A n ) (1 . 1 . 2) . The triple (Ω , F , P ) is called a probability space. Definition 1.1.5. Let (Ω , F , P ) be a probabil- ity space. If a set A ∈ F satisfies P ( A ) = 1, we say that the event A occurs almost surely. 2
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Definition 1.2.1. Let (Ω , F , P ) be a probabil- ity space. A random variable is a real-valued function X
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