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Probability , expectation, variance, cross-term vani

  1. May 11, 2017 #1
    1. The problem statement, all variables and given/known data

    I have a variable ##s_i## with probability distribution ##w(s_i)##
    ##(\Delta(s_i))^2## denotes the variance ##=<(s-<s>)^2>=<s^2>-<s>^2##
    I want to show ## \sum\limits_{i\neq j} <\Delta s_i> < \Delta s_j> =0 ##

    where ## < > ## denote expectation

    My book has:

    ## <\Delta s_i> =\int ds_i w(s_i)(s_i-<s_i>)=0##

    I don't really understand this so the first term gives ##<s_i>## that's fine which would obviously cancel with a ##<s_i>## but isn't the second term ##E(<s_i>)## not ##<s_i>## so how do they cancel?

    Many thanks in advance.


    2. Relevant equations
    see above

    3. The attempt at a solution

    I don't really understand this so the first term gives ##<s_i>## that's fine which would obviously cancel with a ##<s_i>## but isn't the second term ##E(<s_i>)## not ##<s_i>## so how do they cancel?

    Many thanks in advance.
     
  2. jcsd
  3. May 11, 2017 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    ##\langle s_i \rangle## is just a number, so ##\int w(s_i) \langle s_i \rangle \, ds_i = \langle s_i \rangle \int w(s_i) \, ds_i##, and that last integral equals 1.
     
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