In a problem set I proved this "lemma," whose result is not intuitive to me. $Z$ is a standard normal distribution in a censored model.
Formally, $Z^* \sim Norm(0, \sigma^2)$, and $Z = max(Z^*, c)$. Then, \begin{align} E[Z|Z>c] &= \int_c^\infty z_i \phi({z_i})\mathrm{d}z_i \\ &= \frac{1}{\sqrt{2\pi}}\int_c^\infty z_i \exp\!\bigg(\frac{-1}{2}z_i^2\bigg)\mathrm{d}z_i \\ &= \frac{1}{\sqrt{2\pi}} \exp\!\bigg(\frac{-1}{2}c^2\bigg) \quad\quad\quad\quad\text{ (Integration by substitution)}\\ &= \phi(c) \end{align} So there is some sort of connection between the expectation formula over a truncated domain and the density at the point of truncation $(c)$. Could anyone explain the intuition behind this?