I have random variables $(Z,Z_k)\sim N(0,\Sigma)$ and $\bar{y}=\boldsymbol{1}\{Z>-C\}$ for some fixed constant $C$. I would like to compute $\mathbb{E}[Z|Z_k,\bar{y}]$ without using any integration. In what follows, I will outline how I'll do it. We have \begin{align*} \mathbb{E}[Z|Z_k,\bar{y}] &\stackrel{(a)}{=}\mathbb{E}[Z|Z_k,\bar{y},Z>-C]\mathbb{P}[Z>-C|Z_k,\bar{y}] \\ &\qquad+\mathbb{E}[Z|Z_k,\bar{y},Z\leq-C|Z_k,\bar{y}]\mathbb{P}[Z\leq-C|Z_k,\bar{y}] \\ &\stackrel{(b)}{=}\mathbb{E}[Z|Z_k,Z>-C]\boldsymbol{1}\{\bar{y}=1\} +\mathbb{E}[Z|Z_k,Z\leq-C]\boldsymbol{1}\{\bar{y}=0\}, \end{align*} where (a) is by the towering property of expectation, and (b) is by noting that $\bar{y}$ and $Z>-C$ describes the same event. Next, we introduce the definition of a truncated Gaussian random variable. Note that $\boldsymbol{1}\{\cdot\}$ is the indicator function.
Definition: Given $Z$ is a Gaussian random variable, then $(Z|z_l<Z<z_u)$, where $-\infty\leq z_l<z_u\leq\infty$, is a truncated Gaussian random variable with mean \begin{align} \mathbb{E}[Z|z_l<Z<z_u] &=\mathbb{E}[Z]-\sqrt{\text{Var}[Z]}\frac{\phi(\frac{z_u-\mathbb{E}[Z]}{\sqrt{\text{Var}[Z]}})-\phi(\frac{z_l-\mathbb{E}[Z]}{\sqrt{\text{Var}[Z]}})}{\Phi(\frac{z_u-\mathbb{E}[Z]}{\sqrt{\text{Var}[Z]}})-\Phi(\frac{z_l-\mathbb{E}[Z]}{\sqrt{\text{Var}[Z]}})}, \end{align} where $\phi(\cdot)$ is the standard Gaussian pdf and $\Phi(\cdot)$ is the standard Gaussian cdf.
Applying this definition, we have \begin{align} \mathbb{E}[Z|Z_k,\bar{y}] &=\bigg(\mathbb{E}[Z|Z_k]+\sqrt{\text{Var}[Z|Z_k]}\frac{\phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}{1-\Phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}\bigg)\boldsymbol{1}\{\bar{y}=1\} \\ &\qquad+\bigg(\mathbb{E}[Z|Z_k]-\sqrt{\text{Var}[Z|Z_k]}\frac{\phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}{\Phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}\bigg)\boldsymbol{1}\{\bar{y}=0\} \\ &=\mathbb{E}[Z|Z_k]+\sqrt{\text{Var}[Z|Z_k]}\bigg(\frac{\phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}{1-\Phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}\boldsymbol{1}\{\bar{y}=1\}-\frac{\phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}{\Phi(\frac{-C-\mathbb{E}[Z|Z_k]}{\sqrt{\text{Var}[Z|Z_k]}})}\boldsymbol{1}\{\bar{y}=0\}\bigg) \end{align} Is this the correct way to do it? More specifically, does the truncated Gaussian definition transfer over to conditional expectations? Thanks.