This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ddimensional random vector. Here, R is the Borel σ field or the σfield generated by the open subsets of R n , and R d is the dfold product σalgebra 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 σfield Let A be a collection of subsets of Ω. The σfield generated by A , denoted by σ ( A ), is the smallest σfield on Ω which contains A , which is the intersection of all σfields 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 σfield 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...
View
Full Document
 Spring '09
 Reed
 Probability theory, Limit of a function, Supremum, measure, lim inf xn

Click to edit the document details