# hw4 - Homework Set No 4 Probability Theory(235A Fall 2011...

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Homework Set No. 4 – Probability Theory (235A), Fall 2011 Due: 10/25/11 at discussion section 1. 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. (b) Prove Jensen’s inequality , which says that if X is a random variable such that P ( X ( a,b )) = 1 and ϕ : ( a,b ) R is convex, then ϕ ( E X ) E ( ϕ ( X )) . Hint. Start by proving the following property of a convex function: If ϕ is convex then at any point x 0 ( a,b ), ϕ has a supporting line , that is, a linear function y ( x ) = ax + b such that y ( x 0 ) = ϕ ( x 0 ) and such that ϕ ( x ) y ( x ) for all x ( a,b ) (to prove its existence, use the characterization of convexity from part (a) to show that the left-sided derivative of

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hw4 - Homework Set No 4 Probability Theory(235A Fall 2011...

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