Learn parameters for truncated Gaussian I would like to learn the parameters for a truncated gaussian like this one.

I'm using this formula for the probability density
$f(x | \mu, \sigma^2) = \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \times \frac{1}{F(\mu, \sigma)}\times \mathbb{1}_{x>0}$
where $F(\mu, \sigma)$ is a function so that the integral of $f$ over $x$ equals $1$.
I tried to derive the maximum likelihood like we usually do for gaussian mixtures, but then I got stuck : I can't find an easy formula for $\mu$ and $\sigma$. How to find them?
 A: To be clear about what you written in your probability density function, this is a form of truncation. In other words, if $X < 0$, then $X$ does not appear in the data set. As such, it will not be described as a Gaussian Mixture Model. 
When treating this as a maximum likelihood problem, there are no constraints on $\mu$ or $\log(\sigma)$. So vanilla optimization algorithms, like Newton Raphsons, should be able to optimize the likelihood without much problem. 
Alternatively, if you really wanted to fit an EM algorithm, I believe you could do this by having the dropped $X$'s be your missing data. However, this will be a little tricky; your missing data will not just be values of unobserved $X$, but how many $X$'s are missing. The formula for the expected of number missing $X$'s would be $\frac{n}{1 - \Phi(-\mu / \sigma)} - n$. This will not be a whole number with probability 1! 
You should still be able to get a closed form M step by using a weight sample, but this is starting to get considerably more complicated than just using Newton Raphson's at this point. 
