# Variance of a sum of random variables

• May 17th 2014, 08:31 AM
Vinod
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.
• May 17th 2014, 10:09 AM
romsek
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.
• May 17th 2014, 07:08 PM
johng
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:

http://i60.tinypic.com/2l8ym2s.png
• May 18th 2014, 06:55 AM
Vinod
Re: Variance of a sum of random variables
Hi johng,
$\displaystyle \displaystyle\sum_{i=1}^n Cov(X_i,X_i) instead of \displaystyle\sum_{i=1}^n Cov(X_i,X_j)$