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Math Help - Maximizing over a continuum of possibilities

  1. #1
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    Maximizing over a continuum of possibilities

    Let x be a random variable with density function f and [a,b] be the possible set of values of x.

    Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a  C^{\infty} function.

    Let  H(w(x)) = \int_{a}^b f(x)U(w(x))dx

    Find the derivative of H with respect to w over all x.

    I don't know where to begin because I'm not even sure I understand the question.

    I suspect the answer is f(x)U'(w(x)) only because it would be analogous to the case with finitely many possible value of x, but I dont know how to prove that.
    Last edited by southprkfan1; May 9th 2010 at 10:51 AM.
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    Quote Originally Posted by southprkfan1 View Post
    Let x be a random variable with density function f and [a,b] be the possible set of values of x.

    Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a  C^{\infty} function.

    Let  H(w(x)) = \int_{a}^b f(x)U(w(x))dx

    Find the integral of H with respect to w over all x.

    I don't know where to begin because I'm not even sure I understand the question.

    I suspect the answer is f(x)U'(w(x)) only because it would be analogous to the case with finitely many possible value of x, but I dont know how to prove that.
    Are you to find the integral of H or the derivative?

    The derivative is, using the fundamental theorem of calculus and the chain rule, f(x)U'(w(x)) as you say. The integral would be much messier.
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    Quote Originally Posted by HallsofIvy View Post
    Are you to find the integral of H or the derivative?

    The derivative is, using the fundamental theorem of calculus and the chain rule, f(x)U'(w(x)) as you say. The integral would be much messier.
    Oops, Yes I meant the derivative of H. Could you explain how you get the answer in a little more detail.
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    Quote Originally Posted by southprkfan1 View Post
    Oops, Yes I meant the derivative of H. Could you explain how you get the answer in a little more detail.
    Specificially, wouldn't the FTOC imply the answer should be: f(x)U(w(x))*U'(W(x))?
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