This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ( , R , P ) where = [0 , 1], R is the Borel algebra, and P is the Lebesgue measure. Define X by the assignment X ( ) := sup { y : F ( y ) < } 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 f integraldisplay...
View Full
Document
 '07
 RomanVershynin
 Probability

Click to edit the document details