Suppose a random variable $x$ is distributed $\mathcal{N}(\bar{x},\sigma^2)$. I am wondering if there is a way to derive the expectation of this variable multiplied by a Normal CDF: $$\mathbb{E}[x*\Phi(x)]$$ where $\Phi$ is the CDF of a standard Normal random variable. And is $x*\Phi(x)$ a truncated normal r.v.?
2026-04-08 05:59:11.1775627951
Expectation of a Normal r.v. multiplied by a CDF
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