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Unformatted text preview: Homework 4 Probability Theory (MATH 235A, Fall 2007) 1. Generating random variables with given distributions. Consider a funciton F : R → R that satisfies: (i) F is nondecreasing and 0 ≤ F ( x ) ≤ 1 for all x ; (ii) F ( x ) → 0 as x → ∞ and F ( x ) → 1 as x → ∞ ; (iii) F is right continuous. Prove that F is a distrubution function of some random variable X . To do so, consider the probability space (Ω , R , P ) where Ω = [0 , 1], R is the Borel σalgebra, and P is the Lebesgue measure. Define X by the assignment X ( ω ) := sup { y : F ( y ) < ω } and prove that X is a random variable with distribution function F 2. Approximation by simple functions. a) Let f be a bounded measur able function. Prove that there exists a sequence of simple functions f n such that f n → f pointwise. (Hint: see the proof of Lemma 2.1.) b) Refer to Definition 2.5 of Lebesgue integral of simple functions. Prove that for every bounded measurable function f , we have sup φ ≤ f integraldisplay...
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 '07
 RomanVershynin
 Probability, Probability theory, Lebesgue

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