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Thread: proof on expected value inequality

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    proof on expected value inequality




    Suppose f(x) symmetric. Prove E[|x|]^2 <= variance(x) = E[x^2]-E[x]^2.

    Attempt:

    E[|x|]^2 +E[x]^2 <= E[|x|]^2 +E[|x|]^2 = 2E[|x|]^2 (?) <= E[x^2]

    I do not know how to show 2E[|x|]^2 <= E[x^2].
    Last edited by stephanie; 01-24-2014 at 12:44 PM.

  2. #2
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    Re: proof on expected value inequality


    1. Do you know the value of E[X] under the symmetric assumption

    f(x) = f(-x) ?

    2. You may obtain the required inequality by considering Var[|X|] \geq 0,
    or equivalently by Jensen's inequality.

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