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Suppose I have to vectors $w$ and $x$, each of size $[512,1]$. Each element of $w$ and $x$ is an i.i.d sample from a Guassian Distibution with mean 0 and variance 1. So $x_i$ and $w_i$ follow $N(0,1)$

Let $y=w^Tx$ be my vector of interest. Then the question is what is the variance of $y$?

So $y=\sum w_ix_i$

$Var(y)=\sum Var(w_ix_i)$

I do not know how to take it forward from here....

Rahul Deora
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  • The next step is to write the Var operator as an expectation ($Var(X) = E[(X-E[X])^2]$). Then, note that if $A$ and $B$ are independent, then $E[AB]=E[A]E[B]$ – tmrlvi Aug 06 '19 at 19:12
  • Okay so we have $Var(X)=E((AB)^2)$. Then what? – Rahul Deora Aug 07 '19 at 04:03
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    You assume A and B are independent. So also $A^2$ and $B^2$. – tmrlvi Aug 07 '19 at 06:01
  • Ok so I then get $Var(y)=\sum 1*1=n$ . Which is 512 in this case. Suppose W was a $512*512$ matrix. Then $y$ would be a vector where each entry has variance 512. So what would the total variance of vector $y$ be? – Rahul Deora Aug 10 '19 at 15:41

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