Optimal normal shift for normal-tail importance sampling ($2\mu =\phi (z+\mu )/\bar{\Phi }(z+\mu )$)

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 $\phi (a)/\bar{\Phi }(a)=E[Z\mid Z>a]$, the minimizing shift is half the target conditional mean above the shifted threshold. The resulting estimator remains unbiased with the exact likelihood-ratio weight.