pract_final - Stat 310A/Math 230A Theory of Probability...

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Unformatted text preview: Stat 310A/Math 230A Theory of Probability Practice Final Andrea Montanari November 30, 2010 Solutions should be complete and concisely written. Please, mark clearly the beginning and end of each problem. You have 3 hours but you are not required to solve all the problems! Just solve those that you can solve within the time limit. For any clarification on the text, somebody will sit outside the room for all the duration of the exam. You can consult textbooks and your notes. You cannot use computers, and in particular you can not use the web. You can cite theorems from Amir Dembos lecture notes by number, and exercises you have done as homework by number as well. Any other non-elementary statement must be proved! Problem 1 Let = { , 1 } N be the space of infinite binary sequences = ( 1 , 2 , 3 ,... ), and, for a b , write b a for the vector ( a , a +1 ,..., b ). Let F the -algebra gnerated by cylindrical sets C , = : 1 = 1 , (1) for N , . Let P be the product measure over ( , F ), defined by P ( C , ) = Y i =1 p ( i ) , (2) where p (1) = 1- p (0) = p (0 , 1). Define, for (0 , 1 / 2] X ( ) X i =1 i i- 1 , (3) and let P X be its law. (a) Prove that, for = 1 / 2 and any 0 < x 1 < x 2 < 2, P X (( x 1 ,x 2 )) > 0. What happens if (0 , 1 / 2)? (b) Prove that, for (0 , 1 / 2), P X does not have atoms. What happens if = 1 / 2?...
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This note was uploaded on 08/20/2011 for the course STATS 310A at Stanford.

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pract_final - Stat 310A/Math 230A Theory of Probability...

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