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Unformatted text preview: Stat 310A/Math 230A Theory of Probability Midterm Solutions Andrea Montanari November 1, 2010 The midterm was long! This will be taken into account in the grading. We will assign points proportion ally to the number of questions answered (e.g. Problem 1 counts for 4 questions) and then rescale upwards the grades distribution. This is also a good time to discuss any difficulty you encountered with the instructors. Problem 1 Let 2 be the Lebesgue measure on ( R 2 , B R 2 ). We know already that it is invariant under translation i.e. that 2 ( B + x ) = 2 ( B ) for any Borel set B and x R 2 (whereby B + x = { y R 2 : y x B } ). (a) Show that it is invariant under rotations as well, i.e. that for any [0 , 2 ], and any Borel set B R 2 , 2 ( R ( ) B ) = 2 ( B ) (whereby R ( ) denotes a rotation by an angle and R ( ) B = { x R 2 : R ( ) x B } ). Solution : Throughout the solution we will use the fact that 2 = 1 1 whence we obtain the action of 2 on rectangles: 2 ( A 1 A 2 ) = 1 ( A 1 ) 1 ( A 2 ). Also, for J 1 ,J 2 R two intervals, let T J 1 ,J 2 be any triangle with two sides equal to J 1 (parallel to the first axis) and J 2 (equal to the second axis). from the addittivity of 2 it follows immediately that 2 ( T J 1 ,J 2 ) =  J 1   J 2  / 2. (We use here the fact that for a segment S = { x + x 1 : [ a,b ] } , x ,x 1 R 2 , 2 ( S ) = 0, which can be proved by covering S with squares.) Consider next a rectangle A = [0 ,a ) [0 ,b ), and let A = R ( ) A . Using again addittivity (see figure above) it follows that, for = / 2 : 2 ( A ) = ( a cos + b cos )( a sin + b sin ) a 2 sin cos  b 2 sin cos = ab (cos sin + cos sin ) = ab sin( + ) = ab. Hence 2 ( A ) = 2 ( R ( ) A ) and by translation invariance this holds for any A = [ a 1 ,a 2 ) [ b 1 ,b 2 ) (not necessarily with a corner at the origin). Since the system P = { A = [ a 1 ,a 2 ) [ b 1 ,b 2 ) : a 1 < a 2 ,b 1 < b 2 } generates the Borel algebra, and recalling that 2 is finite, this proves the claim by Caratheodory uniqueness theorem. (b) For s R + , and B R 2 Borel, let sB { x R 2 : s 1 x B } . Prove that 2 ( sB ) = s 2 2 ( B )....
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This note was uploaded on 08/20/2011 for the course STATS 310A at Stanford.
 '11
 MONTANARI,A.
 Probability

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