Lecture10

Lecture10 - IID Random Variables Lecture 11 Sequences of...

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•1 Lecture 11 Sequences of IID Random Variables IID Random Variables A great variety of applications require open-ended measurements, which are random in nature Such measurements may not be handled as N- dimensional vectors, but rather as a sequence (i.e., similar to series in calculus) In a course of a large number of experiments, the next trial does not depend on previous ones In addition, the probabilistic behavior of the experiment is invariant from trial to trial, e. g. representing trials as random variables we get that is independent of . i X j X j i } { i X ± Combining these observations yields the following definition Definition 1: A sequence of random variables is called a sequence of independent, identically distributed (IID) random variables if the following conditions are met: 1. 2. For the joint CDF of any finite subsequence of random variables we have +∞ = 1 } { i i X N i R x x F x F i X , ), ( ) ( L k n k X 1 } { = = = L i i L X X x F x x F L n n 1 1 ...... ) ( ) ,... ( 1 Remark Note: The absence of any dependence in a process means that the process has “no memory”, i.e. future is completely independent of the present and of the past

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•2 Sums of IID random variables ¾ Example: In the Dow Jones game, the total gain from an investment strategy is the sum of the daily gains.
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Lecture10 - IID Random Variables Lecture 11 Sequences of...

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