0

Suppose we have n jointly distributed random variables $x_i,i=1,...,n,$ with mean and variance $E(x_i)=\mu_i$, $Var(x_i)=\sigma^2_i$ and covariance $Cov(x_i,x_j)=\sigma_{ij}.$ Then the weighted sum of the n random variable has mean

$E\bigg(\displaystyle\sum_{i=1}^n\omega_ix_i\bigg)=\displaystyle\sum_{i=1}^n \omega_i\mu_i$

and variance

$Var\bigg(\displaystyle\sum_{i=1}^n\omega_ix_i\bigg)=\displaystyle\sum_{i=1}^n\displaystyle\sum_{i=1}^n\omega_i\omega_j\sigma_{ij}$

How to prove both these formulas? Any member may show its correct proof along with examples.

Dhamnekar Winod
  • 806
  • 8
  • 17
  • Please use our site search function: https://stats.stackexchange.com/search?q=covariance+formula+is%3Aanswer+score%3A2. – whuber Apr 13 '19 at 14:48
  • 2
    Duplicate of [Variance of linear combinations of correlated random variables](https://stats.stackexchange.com/questions/160230/variance-of-linear-combinations-of-correlated-random-variables). The formula that you seek is the next-to-last line of the derivation in my inswer to the question that I cite. I refuse to add the examples that you wish for to that answer. – Dilip Sarwate Apr 13 '19 at 14:52

0 Answers0