variance of inner product of vectors sampled from gaussian distribution?

Question

Suppose that we have 2 vectors such W and X $\in R^N$ that each element sampled from normal distribution $\mathcal{N}(0, 1)$, and the inner product $Z=W^T X$.
Why is variance $V[Z]$ equal to $N\sigma^2_{x}(\sigma^2_{w}+\mu^2_{w})$?
Its related with signal propagation in neural network, such that how variance of activation can propagate through layers.

Answer 1

From independence $$E[X_iW_i]=E[X_i]E[W_i]$$

$$E[X_i^2W_i^2]=E[X_i^2]E[W_i^2]=(Var(X_i)+E[X_i]^2)(Var(W_i)+E[W_i]^2)$$

If $E[X_i]=0$, then we have

$$Var(X_iW_i)=Var(X_i)(Var(W_i) + E[W_i]^2)$$

and I am leaving the very last step to you.