rec15_ans - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Recitation 15 April 20, 2006 1. See textbook pg. 399 2. (a) N = 200 , 000. (b) N = 100 , 000. 3. Let us fix some ǫ > 0. We will show that P ( Y n 0 . 5 + ǫ ) converges to 0. By symmetry, this will imply that P ( Y n 0 . 5 ǫ ) also converges to zero, and it will follow that Y n converges to 0.5, in probability. For the event { Y n 0 . 5 + ǫ } to occur, we must have at least n + 1 of the random variables X 1 , X 2 , . . . , X 2 n +1 to have a value of 0 . 5 + ǫ or larger. Let Z i be a Bernoulli random variable which is equal to 1 if and only if X i 0 . 5 + ǫ : 1 if X i 0 . 5 + ǫ Z i = 0 otherwise { Z 1 , Z 2 , .... } are i.i.d random variables and E [ Z i ] = P ( Z i = 1) = P ( X i 0 . 5 + ǫ ) = 0 . 5 ǫ . Hence, for the event { Y n 0 . 5 + ǫ } to occur, we must have at least n + 1 of the { Z i } to take value 1, 2 n +1 P ( Y n 0 . 5 + ǫ ) = P ( Z i n + 1) i =1 2 n +1 Z i n + 1 = P ( i =1 ) 2 n + 1 2 n + 1 2 n +1
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