For simplicity lets assume we have a continuous distribution from 0 to 100.
If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters.
However if the mean is 85 and std is 40, then it is quite clear that the distribution cannot look like a gaussian. My question is, what can we say about such a distribution. How could such a distribution be estimated?
I've attached a picture of various beta densities from the wiki on the Beta Distribution. It really matters what your density looks like beyond the first two moments you specified, but in general, this will ensure the values are restricted to a bounded interval.
If you aren't concerned with some density living outside your interval you might consider using a Gamma Density.
If you really have no idea what the density looks like and want to restrict to a bounded interval you determine the maximal entropy density. This will be the one subject to the constraints of a bounded interval with fixed mean and variance. See this wiki page on the Maximum Entropy Probability Distribution. In particular, the section headed "A theorem due to Boltzmann".