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hw1sol - EE 376A Prof T Weissman Information Theory...

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EE 376A Information Theory Prof. T. Weissman Thursday, January 14, 2010 Homework Set #1 (Due: Thursday, January 21, 2010) 1. Coin flips. A fair coin is flipped until the first head occurs. Let X denote the number of flips required. (a) Find the entropy H ( X ) in bits. The following expressions may be useful: summationdisplay n =0 r n = 1 1 r , summationdisplay n =0 nr n = r (1 r ) 2 . (b) A random variable X is drawn according to this distribution. Find an “efficient” sequence of yes-no questions of the form, “Is X contained in the set S ?” Compare H ( X ) to the expected number of questions required to determine X . Solution: (a) The number X of tosses till the first head appears has the geometric distribution with parameter p = 1 / 2, where P ( X = n ) = pq n - 1 , n ∈ { 1 , 2 ,... } . Hence the entropy of X is H ( X ) = summationdisplay n =1 pq n - 1 log( pq n - 1 ) = bracketleftBigg summationdisplay n =0 pq n log p + summationdisplay n =0 npq n log q bracketrightBigg = p log p 1 q pq log q p 2 = p log p q log q p = H ( p ) /p bits.
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