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

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

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