Say $w_i$ is each constant, $\epsilon_i \sim N(0,\sigma^2)$ and independent of $w_i$, how can one derive the equation $\displaystyle E[(\Sigma_{i=1}^D w_i \epsilon_i)^2]= \Sigma_{i=1}^{D}w_i^2\sigma^2$ ? There's no other assumption on $w_i$
Some of my thought:
I know we can get $E[\epsilon_i^2]=\sigma^2$ and can lead to $E[(\Sigma_{i=1}^D w_i \epsilon_i)^2]=E[\Sigma_{i=1}^{D}w_i)^2] \sigma^2$, but how to make the square be with each $w_i$ instead of outside the sum?
Since $E[\Sigma_{i=1}^{D}w_i)^2] \sigma^2 = E[w_1 + w_2 + ... +w_D)^2]\sigma^2 = E[w_1^2 + w_2^2 + ... +w_D^2 + 2w_1w_2 + 2w_1w_3 + ... + 2 w_{D-1}w_D]\sigma^2$, when will the expectation between those $w_i$ become zero? I learned constant's expectation will just be a constant but not necessarily zero.