= Solution
{c}
An <importance weight> has <expected value> $\mathbb E_q[w(Z)]=\int p(z)\,dz=1$. From the ratio in (a),
$$
\mathbb E_q[w(Z)^2]
=s^d\prod_{j=1}^d\mathbb E_{Z_j\sim N(0,s)}\!\left[e^{-(1-s^{-1})Z_j^2}\right].
$$
The supplied <normal distribution> transform applies with quadratic coefficient $-(1-s^{-1})$, giving each factor $(2s-1)^{-1/2}$. Thus
$$
\boxed{\operatorname{Var}_q(w(Z))=\left(\frac{s}{\sqrt{2s-1}}\right)^d-1.}
$$
For $s>1$, the base $A:=s/\sqrt{2s-1}$ exceeds one because $s^2-(2s-1)=(s-1)^2>0$: <importance weight> <variance> increases exponentially with dimension. For $s=1$, the <importance weight> is identically one and its <variance> is zero. This calculation concerns the <importance weight>, rather than the weighted integrand.
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