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Unformatted text preview: Homework Set No. 4 Probability Theory (235A), Fall 2009 Posted: 10/20/09 Due: 10/27/09 1. If P,Q are two probability measures on a measurable space ( , F ), we say that P is absolutely continuous with respect to Q , and denote this P << Q , if for any A F , if Q ( A ) = 0 then P ( A ) = 0. Prove that P << Q if and only if for any > 0 there exists a > 0 such that if A F and Q ( A ) < then P ( A ) < . For a hint, go to the URL: http://bit.ly/1Nhpkf , but only if you get stuck. 2. A function : ( a,b ) R is called convex if for any x,y ( a,b ) and [0 , 1] we have ( x + (1- ) y ) ( x ) + (1- ) ( y ) . (a) Prove that an equivalent condition for to be convex is that for any x < z < y in ( a,b ) we have ( z )- ( x ) z- x ( y )- ( z ) y- z . Deduce using the mean value theorem that if is twice continuously differentiable and satisfies 00 0 then it is convex....
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