hw2sol

hw2sol - Math 280A Fall 2011 Homework 2 Solutions 1 Let B...

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Unformatted text preview: Math 280A, Fall 2011 Homework 2 Solutions 1. Let B be any Borel subset of R , and compute: { Z ∈ B } = [ A ∩ { Z ∈ B } ] ∪ [ A c ∩ { Z ∈ B } ] = [ A ∩ { X ∈ B } ] ∪ [ A c ∩ { Y ∈ B } ] Since each of the events A , A c , { X ∈ B } , { Y ∈ B } , is an element of F , so is { Z ∈ B } . 2. As n ∈ N increases to ∞ , the events {| X | ≤ n } increase to {| X | < ∞} = Ω. Therefore (continuity of probability) lim n P [ | X | ≤ n ] = 1 . Consequently, given ǫ > 0 we can choose (and fix) a positive integer n ( ǫ ) so large that P [ | X | ≤ n ( ǫ )] ≥ 1 − ǫ . We use this n ( ǫ ) to define X ǫ : X ǫ ( ω ) := braceleftbigg X ( ω ) , if | X ( ω ) | ≤ n ( ǫ ), n ( ǫ ) , if | X ( ω ) | > n ( ǫ ). The function X ǫ so defined is a random variable (by the logic of problem 1). Moreover, | X ǫ ( ω ) | ≤ n ( ǫ ) for all ω ∈ Ω, so that X ǫ is a bounded (by n ( ǫ )) random variable. And finally P [ X negationslash = X ǫ ] ≤ P [ | X | > n...
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This note was uploaded on 12/11/2011 for the course MATH 280a taught by Professor Driver,b during the Fall '08 term at UCSD.

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hw2sol - Math 280A Fall 2011 Homework 2 Solutions 1 Let B...

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