= Solution
{c}
The integrand is positive, so the <minimum-variance importance distribution> is its target-weighted <probability density function>, normalized by $\theta=\sqrt e$:
$$
\boxed{q^\star(z)=\frac{e^{-z_1}p(z)}{\sqrt e}.}
$$
By <completing the square>,
$$
-z_1-\frac12z_1^2-\frac12=-\frac12(z_1+1)^2.
$$
Thus $q^\star$ is the <multivariate normal distribution> $N(-e_1,I_d)$, where $e_1=(1,0,\ldots,0)$ is the first <standard basis> vector. Its <importance weight> and weighted integrand are
$$
\boxed{w^\star(z)=e^{z_1+1/2},\qquad
w^\star(z)e^{-z_1}=\sqrt e,\qquad
\min_q\operatorname{Var}(\widehat\theta_n^{\mathrm{IS}})=0.}
$$
A <variance> cannot be negative, and this proposal makes every summand constant, proving optimality directly. It is admissible under the arbitrary-proposal condition even though its <mean vector> is not zero. Although optimal proposals usually involve an unknown normalizing integral, here both that integral and the resulting <normal distribution> are explicit.
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