= Solution
For the weighted loss, the <Bayes estimator under parameter-weighted squared error> is the mean of the posterior after multiplication by
$$
\theta^{\alpha-1}(1-\theta)^{\beta-1}.
$$
Since the original posterior density is proportional to
$$
\theta^X(1-\theta)^{n-X},
$$
the reweighted density is $\operatorname{Beta}(X+\alpha,n-X+\beta)$. Its mean gives
$$
\widetilde\theta=\frac{X+\alpha}{n+\alpha+\beta}.
$$
Writing this in the required form,
$$
\boxed{
\widetilde\theta
=w\widehat\theta+(1-w)\theta_0,
\qquad
w=\frac n{n+\alpha+\beta},
\qquad
\theta_0=\frac{\alpha}{\alpha+\beta}
}.
$$
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