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Homework Help: Proof that's missing a step

  1. Nov 27, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that

    c(0) = [itex]\sigma[/itex][itex]^{2}[/itex]

    2. Relevant equations

    c([itex]\ell[/itex]) = [itex]\frac{1}{N-\ell}[/itex] [itex]\sum[/itex] y[itex]_{i}[/itex]y[itex]_{i+\ell}[/itex], with the summation going from i=1 to N-[itex]\ell[/itex]

    y(i) = x(i)-[itex]\langle[/itex]x[itex]\rangle[/itex]

    [itex]\langle[/itex]x[itex]\rangle[/itex]=[itex]\frac{1}{N}[/itex][itex]\sum[/itex]x(i)

    [itex]\langle[/itex]x2[itex]\rangle[/itex]=[itex]\frac{1}{N}[/itex][itex]\sum[/itex]x(i)2

    Edit: the other summations go from 1 to N

    Edit: forgot sigma

    [itex]\sigma[/itex]2 = ([itex]\langle[/itex]x2[itex]\rangle[/itex]-[itex]\langle[/itex]x[itex]\rangle[/itex])/(N-1)

    3. The attempt at a solution

    c(0) = [itex]\frac{1}{N}[/itex][itex]\sum[/itex]y(i)y(i)
    = [itex]\frac{1}{N}[/itex][itex]\sum[/itex](x(i)2-2x(i)[itex]\langle[/itex]x[itex]\rangle[/itex]+[itex]\langle[/itex]x[itex]\rangle[/itex]2)
    = [itex]\langle[/itex]x2[itex]\rangle[/itex]-2[itex]\langle[/itex]x[itex]\rangle[/itex]2+[itex]\langle[/itex]x[itex]\rangle[/itex]2
    = [itex]\langle[/itex]x2[itex]\rangle[/itex]-[itex]\langle[/itex]x[itex]\rangle[/itex]2



    Where did I mess up?
     
    Last edited: Nov 27, 2011
  2. jcsd
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