Lecture3

# Lecture3 - Lecture 3 Random variables and their...

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Unformatted text preview: Lecture 3 : Random variables and their distributions 3.1 Random variables Let (Ω , F ) and ( S, S ) be two measurable spaces. A map X : Ω → S is measurable or a random variable (denoted r.v.) if X- 1 ( A ) ≡ { ω : X ( ω ) ∈ A } ∈ F for all A ∈ S One can write { X ∈ A } or ( X ∈ A ) as a shorthand for { ω : X ( ω ) ∈ A } = X- 1 ( A ). If ( S, S ) = ( R d , R d ), then X is called a d-dimensional random vector. Here, R is the Borel σ- field or the σ-field generated by the open subsets of R n , and R d is the d-fold product σ-algebra of R with itself, which will be defined shortly. Perhaps the simplest example of a measurable function is an indicator function of a measurable set. The indicator function of a set F ∈ F is defined as 1 F ( ω ) = ( 1 if ω ∈ F 0 if ω / ∈ F 3.2 Generation of σ-field Let A be a collection of subsets of Ω. The σ-field generated by A , denoted by σ ( A ), is the smallest σ-field on Ω which contains A , which is the intersection of all σ-fields containing A . Let { X i } i ∈I be a family of mappings of Ω into measurable spaces ( S i , S i ), i ∈ I . Here, I 6 = ∅ is an arbitrary (possibly uncountable) index set . The σ-field generated by { X i } i ∈I , denoted by σ ( { X i } i ∈I ), is the smallest σ- field on Ω with respect to which each X i is measurable. Taking A = S i ∈I { X- 1 i ( S ) : S ∈ S i } , we can see that this case...
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Lecture3 - Lecture 3 Random variables and their...

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