Im looking for the conditional mean and variance of a normal distribution given its bigger than $0$.
I hope its clear but in the question, first they are drawings from a $\mathcal N(\mu,\sigma^2)$ distribution. Then all the draws which are greater than $0$ are taken and from that drawing the mean and variance has to be calculated.
I thought that the CDF and PDF of those functions are: $$ F_X(x) = P(X<x) = \frac{\Phi(\frac{x-\mu}{\sigma})-\Phi(\frac{0-\mu}{\sigma})}{1-\Phi(\frac{0-\mu}{\sigma})} $$ $$ f_X(x) = \frac{\phi(\frac{x-\mu}{\sigma})}{\sigma(1-\Phi(\frac{0-\mu}{\sigma}))} $$ with $x \in [0,1]$. and $\Phi$ and $\phi$ are the standard normal cdf and pdf.
Then the mean should be calculated as: $$ \mu = \int_{0}^{\infty}x\frac{\phi(\frac{x-\mu}{\sigma})}{(1-\Phi(\frac{0-\mu}{\sigma}))\sigma}dx $$ and the variance with $x^2$ in front of the fraction. But i cant figure out how to solve that equation. hope you can help me.