Say $X \sim N \left( \mu, \sigma \right)$. Now I have another random variable $Y$ which is created by retaining all positive values of $X$ as original but all negative values of $X$ are censored at 0.
I am wondering if there is any analytical closed form formula to calculate the value of $E \left[ Y\right]$.
I searched over internet and stack exchange as well, but so far I am only able to find similar discussion on Truncated random variables not censored one.
Any pointer will be very helpful.
We are trying to find the expectation of $Y = \max(0,X)$. Conditional on the event $\{ X \leq 0 \}$ we have $Y=0$, whereas on the event $\{ X \geq 0 \}$, $Y$ has the distribution of a truncated $\mathcal{N}(\mu,\sigma^2)$ distribution on the interval $(0,\infty)$.
The truncated normal distribution has mean $\mu + \sigma \phi(\mu/\sigma)/\Phi(\mu/\sigma)$, where $\phi$ is the standard normal PDF, and the probability that $X > 0$ is $\Phi(\mu/\sigma)$, where $$\Phi(x) = \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz$$ is the standard normal CDF. Hence by the law of total expectation, $$\mathbb{E}[Y] = \Phi \left (\frac{\mu}{\sigma} \right )\left (\mu + \sigma \frac{\phi(\mu/\sigma)}{\Phi(\mu/\sigma)} \right).$$