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.

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.

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

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 \displaystyle\sum_{i=1}^n Cov(X_i,X_i) instead of \displaystyle\sum_{i=1}^n Cov(X_i,X_j) $