In one paper authors tell:

We let the initialised elements in $W_l$ be independent and identically distributed (i.i.d.). We assume that the elements in $x_l$ are also i.i.d. and $x_l$ and $W_l$ are independent of each other. Then we have:


We let $w_l$ have zero mean. Then the variance of the product of independent variables gives us:

$$Var[y_l] = n_lVar[w_l]E[x_l²]$$

Could someone please describe what might be the way from the first equation to the second?


$$\begin{align}\operatorname{Var}(w_lx_l)&=\mathbb E[w_l^2x_l^2]-\mathbb E[w_lx_l]^2=\mathbb E[w_l^2]\mathbb E[x_l^2]-\mathbb E[w_l]^2\mathbb E[x_l]^2=\mathbb E[w_l^2]\mathbb E[x_l^2]\\&=\operatorname{Var}(w_l)\mathbb E[x_l^2]\end{align}$$

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