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How to expand the average <(N - <N>)^2>

  1. Oct 29, 2012 #1
    Basically I would like to know how to expand:

    [itex]\left\langle[/itex](N - [itex]\left\langle[/itex] N [itex]\right\rangle[/itex])2[itex]\right\rangle[/itex]

    <(N - <N>)^2>

    Where < and > represent [itex]\left\rangle\right\langle[/itex] and denote the average of the quantity the enclose.

    So this is pretty much the average of [N - N(average)] 2

    thank you
  2. jcsd
  3. Oct 29, 2012 #2
    Well first expand the square, and the average of three terms you'll get is the sum of their averages.

    And note that the average of a constant is the same constant.
  4. Oct 30, 2012 #3
    So by expanding i get:

    <N2-2N<N> - <N>2>

    and you're saying this is equivalent to:

    <N2> - <2N<N>> - <<N>2>

    is that right?
  5. Oct 31, 2012 #4
    Yes except the last term should be positive.
  6. Nov 1, 2012 #5


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    Staff: Mentor

    You can simplify <2N<N>> and <<N>2> and combine them to get a nice, short result.
  7. Nov 1, 2012 #6
    Yea I figured that, I simplified

    <N2> - 2N<N> + <N>2


    <N2> - 2<N>2 + <N>2

    so I get

    <N2> - <N>2

    Also I know that <N>2 = <N> + <N>2

    Is there a simple way of showing that without resulting to a probability with sums of exponentials etc?
  8. Nov 2, 2012 #7


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    Staff: Mentor

    I think that equation has an error.

    Your expression is simply the variance: <N2> - <N>2 = Var(N)
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