# Module3_1 - Module 3 Lecture 1 The Asymptotic Equipartition...

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Module 3, Lecture 1 The Asymptotic Equipartition Property G.L. Heileman Module 3, Lecture 1

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Asymptotic Equipartition Property Consider an experiment in which you tossed a fair coin 100 times. Would you be surprised if you observed 100 heads in a row? Well, since the process is iid, the sequence of 100 heads in a row has the same probability as any other 100-element sequence. Intuitively, however, we would know that something is not “right” if we tossed 100 heads in a row. We expect to see about the same number of heads and tails in the sequence if the coin is fair. What we need is the right probabilistic tool for analyzing this situation — something that captures the idea that the number of heads and tails should be about the same. G.L. Heileman Module 3, Lecture 1
Asymptotic Equipartition Property Let X 1 , . . . , X n by iid p ( x ). If p 1 , . . . , p |X| are the probabilities associated with the outcomes of this discrete RV, then if we consider a long sequence (i.e., n large), it should contain with high probability p 1 n occurrences of the first outcome, p 2 n occurrences of the second outcome, etc. Thus, the probability of a particular (long) sequence should be roughly equal to p p 1 n 1 · p p 2 n 2 · · · p p |X| n |X| . Ex. Consider X = { 0 , 1 } , p = Pr { X = 1 } , q = Pr { X = 0 } = 1 - p . Then the sequence 10011011 has probability p 5 q 3 . Furthermore, we expect that a long sequence of length n drawn according to this distribution will contain roughly p · n 1s and (1 - p ) · n 0s. G.L. Heileman Module 3, Lecture 1

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Asymptotic Equipartition Property The asymptotic equipartition property (AEP) relates the asymptotic behavior of iid sequences to entropy. We will use the AEP to divide the set of all sequences derived from an iid process into two sets, those that are typical, and those that are not. We will show that almost all of the probability is associated with the elements in the typical set. Thus, if we prove some property about the typical set, it will apply with high probability to any long sequence drawn according to the iid process. In other words, we will be proving something about the expected behavior of a long sequence. In this module, we will show how the AEP is related to source coding. We will later extend the AEP in order to prove the joint AEP for certain “joint” sequences of RVs, and use this result to prove important results in channel coding.
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