= Optimal normal shift for normal-tail importance sampling
{c}
{title2=$2\mu=\varphi(z+\mu)/\bar\Phi(z+\mu)$}
To estimate $\mathbb P(Z>z)$ for a standard normal variable using proposal $N(\mu,1)$, the weighted indicator has second moment $e^{\mu^2}\bar\Phi(z+\mu)$. Differentiating it at a minimum gives the displayed equation. Since $\varphi(a)/\bar\Phi(a)=E[Z\mid Z>a]$, the minimizing shift is half the target conditional mean above the shifted threshold. The resulting estimator remains unbiased with the exact likelihood-ratio weight.
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