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Math Help - Variance of a sum of random variables

  1. #1
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    Variance of a sum of random variables

    Hi Members,
    Var $\left(\displaystyle\sum_{i=1}^n X_i\right) =Cov \left(\displaystyle\sum_{i=1}^n X_i,\displaystyle\sum_{j=1}^n X_j\right)$
    =$\displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n Cov(X_i,X_j)$

    =$\displaystyle\sum_{i=1}^n Cov(X_i,X_j) +\displaystyle\sum_{i}\displaystyle\sum_{j\not=i}C ov(X_i,X_j)$
    =$\displaystyle\sum_{i=1}^n Var(X_i) + 2\displaystyle\sum_{i}\displaystyle\sum_{j<1} Cov(X_i,X_j)$
    Now, in the last step, from where 2 arise? Can any member suggest me.
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  2. #2
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    Re: Variance of a sum of random variables

    it just uses the fact that $Cov(x,y)=Cov(y,x)$ and redoes the series indices to only count each one once.
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  3. #3
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    Re: Variance of a sum of random variables

    Hi,
    I think a direct derivation of the formula is easier than trying to use properties of covariance:

    Thanks from Vinod
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  4. #4
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    Re: Variance of a sum of random variables

    Hi johng,
    Your reply satisfied my query.BTW, in my thread, please read the second last step as
     \displaystyle\sum_{i=1}^n Cov(X_i,X_i) instead of \displaystyle\sum_{i=1}^n Cov(X_i,X_j)
    Last edited by Vinod; May 18th 2014 at 07:04 AM.
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