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Let X be a continuous random variable. What value of b minimizes E (|X-b|)? Giv

  1. Sep 4, 2011 #1
    Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Giv

    1. The problem statement, all variables and given/known data

    Let X be a continuous random variable. What value of b minimizes E(|X-b|)? Give the derivation


    3. The attempt at a solution

    E(|X - b|)

    E[e - [itex]\bar{x}[/itex]] = E(X)

    E(|E[e - [itex]\bar{x}[/itex]] - b|)

    so ?,.... 0 = E(|E[e - [itex]\bar{x}[/itex]] - E|)

    but this is a graduate course, I have a funny feeling that I am supposed to derive this using a the integral of an Expected value.
     
  2. jcsd
  3. Sep 4, 2011 #2

    lanedance

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    Re: Let X be a continuous random variable. What value of b minimizes E(|X-b|)?

    what is e? Its not clear what your steps are attempting

    I think the integral would be a good way to approach this
     
  4. Sep 4, 2011 #3
    Re: Let X be a continuous random variable. What value of b minimizes E(|X-b|)?

    The e is supposed to be an observation in the sample set
     
  5. Sep 4, 2011 #4

    lanedance

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    Re: Let X be a continuous random variable. What value of b minimizes E(|X-b|)?

    ok well its still not real clear what you're trying to do

    i would try and write the expectation in integral form and consider differentiating, though you may need to be careful with the absolute value
    [tex]f(b) = E[|X-b|] = \int_{-\infty}^{\infty} dx.p(x).|x-b| [/tex]
     
    Last edited: Sep 5, 2011
  6. Sep 4, 2011 #5

    lanedance

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    Re: Let X be a continuous random variable. What value of b minimizes E(|X-b|)?

    if the absolute value sign gives you trouble, you could consider using b to break up the integral into a sum of two integrals (x<b and x>b), this however will complicate the differentiation as now b appears in the integration limit also
     
    Last edited: Sep 5, 2011
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