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Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X₁, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) 3 and (X1, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞

Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X₁, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) 3 and (X1, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.
Transcribed Image Text:2. Let X₁, X2, X3,..., be independent identically distributed Unif(0, 1) random variables. Let Sn be the largest gap between any two of the points {X1, X2,..., Xn} (e.g. if n = 3 and (X₁, X2, X3) = (0.1, 0.5, 0.7) then S3 = 0.4). Show that for any x > 0 lim P(nSn > x) = 1. n→∞ Hint: to lower bound nSn > x, if suffices to demonstrate that there is an empty interval of some length x/n which contains none of the points. Make a list of some candidate intervals and then use Cheby- shev's inequality to show one of them is empty with probability going to 1.
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