# Sum of product as product of sums

Assuming I have two random non-independent vectors $$A,B$$ which are within [-1,1]. I want to approximate their sum of product by product of sums (everything is a dot product), i.e.

$$\sum_{i=1}^NA_iB_i \approx \sum_{i=1}^NA_i \sum_{i=1}^N B_i$$

I can see that this approximation is more and more better if the covariance between the two is closer to zero from the following formula

$$Cov(XY) = E(XY) - E(X)E(Y)$$

How do I achieve this without destroying any signal in $$A,B$$? I am only interested in sample estimate of covariance, i.e. even if reducing the sample size or centering and scaling helps to reduce the absolute value of covariance, it will work.

You arrived at the definition of independence of random variables on your own, congratulations! One formulation is $$E[AB]=E[A]E[B]$$ This is true when variables are independent of each other (not the other way around though).
• perturbation is from Normal distribution $N(0,\sigma^2)$, where $\sigma$ is always >0. Goal of perturbing is to make the vectors noisy and see how the noise impacts the model outcome (model is a neural network). – lvdp Feb 11 '19 at 15:38