Given $X \sim N\left(\mu, \sigma^2 \right)$, I want to find the distribution of
$$Y = \Phi \left( X \right),$$
with $\Phi \left(x\right) = \int_{-\infty}^{x} \frac{1}{ \sqrt{2\pi}} e^{-\frac{t^2}{2}}dt$, i.e., CDF
of a Standard Normal Distribution
.
More specifically, I want to get an expression for $SD \left[Y \right]$, the standard deviation of $Y$.
For the specific case with $\mu = 0, \sigma=1$, I would easily have $Y \sim U\left(0,1 \right)$, but what will be the case for any normal distribution
?
Any pointers will be very helpful.
CDF
of Standard Normal Distribution. I have updated my original post with this information $\endgroup$