= Solution
{c}
Assume the necessary <support condition for importance sampling>, $p\ll q$; the source's density assumptions alone do not ensure it. Put $\mu=E_p f$ and $w=p/q=(Z_q/Z_p)\tilde p/\tilde q$. The <importance sampling> estimator is
$$
\boxed{\hat\mu_N=\frac1N\sum_{i=1}^Nw(X_i)f(X_i).}
$$
Integration under $q$ gives $E_q[wf]=\int pf=\mu$, proving unbiasedness. Independence gives
$$
\boxed{\operatorname{Var}(\hat\mu_N)=\frac1N
\left(\int\frac{p(x)^2f(x)^2}{q(x)}\,dx-\mu^2\right).}
$$
This variance is finite precisely when the first integral is finite. The assumption $E_pf^2<\infty$ ensures target square integrability, but need not ensure square integrability after importance reweighting; the variance can be infinite.
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