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Can someone help me with the proof here? How do I start the proof? How do I simplify $(\sum_{i=1}^k a_i(Y_i - E(Y_i))^2$?

I'd end up getting a large multiplication between each $a_i(Y_i - E(Y_i)$ making my calculation really long.

Are there any summation properties that may help me as I am probably unaware of them?

Thank you!

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Be brave!

$$\begin{align}E\left[\left(\sum_{i=1}^k a_i(Y_i - E[Y_i])\right)^2\right] &= E\left[\sum_{i=1}^k \sum_{j=1}^k a_i a_j (Y_i - E[Y_i])(Y_j - E[Y_j])\right] \\ &= \sum_{i=1}^k \sum_{j=1}^k a_i a_j E\left[(Y_i - E[Y_i])(Y_j - E[Y_j])\right]. \end{align}$$

Now, what can you say about $E\left[(Y_i - E[Y_i])(Y_j - E[Y_j])\right]$ when $i \ne j$? (Hint: use independence.)

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