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hw4solutions - Solutions to Homework Set No 4 Probability...

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Solutions to Homework Set No. 4 – Probability Theory (235A), Fall 2011 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 . (1) Deduce using the mean value theorem that if ϕ is twice continuously differentiable and satisfies ϕ 00 0 then it is convex. Solution. Let x z y and α = ( y - z ) / ( y - x ) so that z = αx + (1 - α ) y and α [0 , 1]. If ϕ is convex, then ϕ ( z ) y - z y - x ϕ ( x ) + z - x y - x ϕ ( y ) , subtracting ϕ ( z )( z - x ) / ( y - x ) from both sides, ϕ ( z ) - ϕ ( x ) z - x ϕ ( y ) - ϕ ( z ) y - z for any x z y Reverse process gives the original condition. For the second part of question, use the mean value theorem for [ x, z ] and [ z, y ] and ϕ 00 0 to give (1). (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 ϕ at x 0 is less than or equal to the right-sided derivative at x 0 ; the supporting line is a line 1
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passing through the point ( x 0 , ϕ ( x 0 )) whose slope lies between these two numbers). Now take the supporting line function at x 0 = E X and see what happens.
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