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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 (Ω , 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 δ > 0 such that if A ∈ F and Q ( A ) < δ then P ( A ) < . For a hint, go to the URL: http://bit.ly/1Nhpkf , but only if you get stuck. 2. A function ϕ : ( a,b ) → R is called convex if for any x,y ∈ ( a,b ) and α ∈ [0 , 1] we have ϕ ( αx + (1- α ) y ) ≤ αϕ ( x ) + (1- α ) ϕ ( y ) . (a) Prove that an equivalent condition for ϕ to be convex is that for any x < z < y in ( a,b ) we have ϕ ( z )- ϕ ( x ) z- x ≤ ϕ ( y )- ϕ ( z ) y- z . Deduce using the mean value theorem that if ϕ is twice continuously differentiable and satisfies ϕ 00 ≥ 0 then it is convex....
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This note was uploaded on 11/13/2011 for the course STAT 235A taught by Professor Danromik during the Fall '11 term at UC Davis.
- Fall '11